|
69d6d14f |
Prove the zero set of A383733 (3-colorings of chorded cycles $C_n^{(3)}$) is exactly $\{7, 8, 12, 16\}$ |
ACTIVE |
1 inv |
2.0 |
4.0 |
23d ago |
|
456c1f41 |
Does Barker's conjectured order-10 recurrence for A321614 (maximum kings on a $4\times 2n$ board, free count) hold beyond the 22-term b-file? |
ACTIVE |
1 inv |
2.0 |
5.0 |
23d ago |
|
0571ec8b |
Density of non-representable sums of $p^kq^l$ with no divisibility, for $\{p,q\}\neq\{2,3\}$ (Erdős #1110) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
74e5240d |
Is there a slowly growing 'good' pairwise-coprime sieving sequence? (Erdős #1101) |
OPEN |
0 inv |
2.0 |
2.0 |
29d ago |
|
b5df427f |
Estimate $f(k)$: the longest run of $k$-smooth consecutive integers above $k$ (Erdős #961) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
756dc791 |
Bound the powerful part $Q_2$ of a product of consecutive integers (Erdős #935) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
6f503dbd |
Finitely many pairs of consecutive-integer blocks (lengths ≥3) with identical prime support? (Erdős #931) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
7336536c |
Estimate h(n): shortest interval holding distinct multiples of each of the first π(n) primes (Erdős #860) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
65b95cb8 |
Are there infinitely many n whose totient valence g(n)=#{m:φ(m)=n} exceeds n^{1−ε}? (Erdős #821) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
6a47bb11 |
Can every integer N≥2 be written as a ratio of two products of consecutive integers? (Erdős #686) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
30743bd5 |
Are there infinitely many n with ω(n−k) < (1+ε)·log k/log log k for all large k? (Erdős #679) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
e0177763 |
Largest LCM-triple-free subset of $\{1,\ldots,N\}$: estimate $f(N)$; is $f(N)=o(N)$? (Erdős #536) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
5c6fd09b |
Almost-sure real-root count of random $\pm1$ polynomials: is $R_n/\log n\to 2/\pi$? (Erdős #521) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
f8a5c1e2 |
Must the survivors of a general congruence sieve have a logarithmic density? (Erdős #486) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
fbd9f7f5 |
Ostmann's inverse Goldbach problem: can $A+B$ be the primes up to finitely many exceptions? (Erdős #431) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
33258de2 |
Irrationality of $\sum a_n/2^{a_n}$ for increasing integer sequences with $a_n/n\to\infty$ (Erdős #260) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
8b197be0 |
$K_{\aleph_1}$-free graphs forcing a monochromatic $K_{\aleph_0}$ under every countable edge-colouring (Erdős #1174) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
5374bcec |
Is $\omega_1^2\not\to(\omega_1^2,k)^2$ provable in ZFC for every finite $k$? (Erdős #1169) |
OPEN |
0 inv |
2.5 |
1.0 |
29d ago |
|
9a44b3c9 |
Does chromatic number $\mathfrak{m}$ force a subgraph of every smaller infinite chromatic number? (Erdős #739) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
f9782c10 |
Do the finite subgraphs of one $\aleph_1$-chromatic graph realise every chromatic number? (Erdős #736) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
a1c89f74 |
For which set-theoretic hypotheses does $2^{\aleph_0}\not\to[\aleph_1]^2_3$ hold? (Erdős #474, $100) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
0fafeb6a |
Edge-colouring an $\aleph_1$-chromatic graph so every countable vertex colouring meets all edge colours (Erdős #1176) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
75424ece |
A cluster of Erdős–Hajnal partition relations at $\omega_2$ and $\omega_3$ under GCH (Erdős #1172) |
OPEN |
0 inv |
2.5 |
1.0 |
29d ago |
|
f421c041 |
Does $\omega_1^2\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$ hold for every finite $k$? (Erdős #1171) |
OPEN |
0 inv |
2.0 |
1.0 |
29d ago |
|
1fc09502 |
Consistency of the symmetric partition relation $\omega_2\to(\alpha)^2_2$ for all $\alpha<\omega_2$ (Erdős #1170) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
a4415c5e |
Does every $\alpha\in[0,1]$ arise as the Hausdorff dimension of a subring or subfield of $\mathbb{R}$? (Erdős #1154) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
29c2dc64 |
Must a finite-subset choice function on a set of size $\aleph_\omega$ admit an infinite independent set? (Erdős #623) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
0d3dd88b |
Infinite sets with $\le 2$ representations of each $n$: is $\liminf|A\cap[1,N]|/N^{1/2}=0$? (Erdős #158) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
860fcc10 |
Does the mean-square gap of the sumset of a finite Sidon set tend to infinity? (Erdős #153) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
472a8e18 |
Prove $\aleph_{\omega+1}\not\to(\aleph_{\omega+1},3,\ldots,3)^2_{\aleph_0}$ in ZFC without GCH (Erdős #1168) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
f2398d7b |
Does $2^\lambda\to(\kappa_\alpha+1)^{r+1}$ imply $\lambda\to(\kappa_\alpha)^r$? (Erdős #1167) |
OPEN |
0 inv |
2.5 |
1.0 |
29d ago |
|
a591ccfb |
Avoiding a sum-free set: a continuum-size $A$ with $A+A$ disjoint from $S$? (Erdős #949) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
2b217698 |
Colour the countable subsets of a cardinal so every $\kappa$-sized set is polychromatic (Erdős #598) |
OPEN |
0 inv |
2.0 |
1.0 |
29d ago |
|
4abfef18 |
Which countable ordinals are partition ordinals: when is $\omega^\beta\to(\omega^\beta,3)^2$? (Erdős #592) |
OPEN |
0 inv |
4.0 |
1.0 |
29d ago |
|
d7df8c65 |
Largest subset of $\{1,\ldots,N\}$ with no two elements whose sum divides their product (Erdős #327) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
5e41787c |
Maximum size of a minimally-vanishing signed unit-fraction set in $\{1,\ldots,N\}$ (Erdős #319) |
OPEN |
0 inv |
2.0 |
2.0 |
29d ago |
|
25c62048 |
Must an infinite real set with $\lvert kx-y\rvert\geq 1$ for all pairs and all $k\geq 1$ be sparse? (Erdős #143) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
0830dac3 |
Minimal non-zero signed reciprocal sum Σ δ_k/k with δ_k ∈ {−1,0,1}: how small can it be? (Erdős #317) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
688830a6 |
Are there infinitely many primary pseudoperfect numbers: 1/p_1+…+1/p_k = 1 − 1/m? (Erdős #313) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
905df96a |
Can a sub-sum of reciprocals approach 1 from below within e^{-cK} once the mass exceeds K? (Erdős #312) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
d07f2072 |
Closest a distinct-unit-fraction sub-sum can get to 1: is δ(N) = e^{-(c+o(1))N}? (Erdős #311) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
49a51261 |
Represent every a/b (b squarefree) as a sum of distinct 1/(pq) with p,q distinct primes (Erdős #306) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
619bae4c |
Longest shortest Egyptian-fraction expansion: estimate N(b), is N(b) ≪ log log b? (Erdős #304) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
f5dd72db |
Largest subset of {1,…,N} with no 1/a = 1/b + 1/c: estimate f(N) (Erdős #302) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
a69b2f1f |
Largest subset of {1,…,N} with no 1/a equal to a sum of distinct 1/b_i: estimate f(N) (Erdős #301) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
4ac8f68c |
Do the first $N$ cubes contain a Sidon set of size $\gg N$? (Erdős #1206) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
c48e9d1c |
Largest Sidon subset guaranteed in every N-point real set: is $\ell(N)\sim N^{1/2}$? (Erdős #530) |
ACTIVE |
1 inv |
3.0 |
2.0 |
18d ago |
|
00af2f59 |
Largest subset of {1,...,N} with all pairwise products distinct: pin the constant in $F(N)$ (Erdős #425) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
e9c8aed9 |
Does $k(N)-(e-1)N\to\infty$? Terms needed for a unit-fraction sum to $1$ with denominators $\geq N$ (Erdős #295) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
74ee34bf |
Growth of $v(k)$, the least integer missing from every $k$-term unit-fraction representation of $1$ (Erdős #293) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
dbce7ae6 |
For all large $k$, can $1$ be written as a sum of reciprocals over $k$ disjoint integer intervals? (Erdős #289) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
bf0af53b |
Are there only finitely many pairs of integer intervals whose reciprocal sums total an integer? (Erdős #288) |
OPEN |
0 inv |
2.0 |
3.5 |
29d ago |
|
ead15314 |
Does the odd-greedy Egyptian-fraction algorithm always terminate for odd-denominator rationals? (Erdős #282) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
9077a647 |
Is there an infinite composite-coordinate path in the visible-lattice-point graph? (Erdős #1212) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
6c54dfc0 |
Do most integers n have a large prime factor within a bounded window n,...,n+k? (Erdős #1201) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
26eff08f |
Can primes of bounded reciprocal sum cover every integer below x by congruences? (Erdős #1200) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
9bd810f8 |
Is the completely-multiplicative random partial sum a.s. unbounded relative to N^{1/2}? (Erdős #1144) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
a8c2db46 |
Which sequences b_n admit a primitive sequence a_n growing no faster than b_n? (Erdős #892) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
7d0410c7 |
How long can the primitive-set saturation game be forced to last? (Erdős #872) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
7ef01369 |
Estimate the Folkman numbers F(k): a monochromatic k-set with all subset sums one colour (Erdős #531) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
a21d6917 |
Does the Rademacher random multiplicative partial sum obey an iterated-logarithm law? (Erdős #520) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
3dcfcd6f |
Is $f(n,k)=(1-\rho(\alpha)+o(1))k$ for the count of $n+i$ with prime factor $>k$? (Erdős #1184) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
66bd02c7 |
Estimate $F_k(p_1,\ldots,p_u)$: multiples of some $p_i$ forced in every length-$k$ interval (Erdős #1143) |
OPEN |
0 inv |
2.0 |
2.5 |
29d ago |
|
262a2c1a |
Integers $n>105$ with $n-2^k$ prime for all $1<2^k<n$: any, or infinitely many? (Erdős #1142) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
adc11e2a |
Gaps between integers with at most two prime factors: is $\limsup (u_{k+1}-u_k)/\log k=\infty$? (Erdős #1139) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
07a1e5a7 |
Infinitely many primes $p$ with every $p-k!$ composite (for $k!<p$)? (Erdős #1059) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
f7f07f6f |
Erdős–Selfridge prime classes: infinitely many primes per class, and growth of $p_r^{1/r}$ (Erdős #1055) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
aca7fd16 |
Growth of $f(n)=\sum_{p<n}1/(n-p)$: liminf, limsup, and an $o(\log\log n)$ bound (Erdős #950) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
4ca68a54 |
Restricted prime-factor counts over consecutive integers: a liminf bound and a limsup law (Erdős #890) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
3f2bb9fd |
Smallest even value missing from the first $x$ prime gaps: does $r(x)\to\infty$? (Erdős #853) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
cb9bf76e |
Longest run of distinct consecutive prime gaps: estimate $h(x)$ (Erdős #852) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
378c569f |
Is there a composite $n+k$ with least prime factor $p(n+k)>k^2$ for all large $n$? (Erdős #681) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
346a8881 |
Least prime factor spikes: is $p(n+k)>k^2+1$ solvable for every large $n$? (Erdős #680) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
54a1b295 |
Is there $f(n)\to\infty$ with a composite $m$ satisfying $n+f(n)<m<n+p(m)$? (Erdős #463) |
OPEN |
0 inv |
2.0 |
2.5 |
29d ago |
|
3b24ada0 |
Is the least-prime-factor sum $\sum p(n)/n$ over every short window $\gg 1$? (Erdős #462) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
71b06748 |
Distinct $t$-smooth components in a window of length $t$: is $f(n,t)\gg t$? (Erdős #461) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
67afa874 |
A positive relative-density set $A$ with all $n-a$ prime for infinitely many $n$ (Erdős #428) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
9bf3a6ac |
Does $\{p+\lfloor C^k\rfloor\}$ have positive density for every $C>1$? (Erdős #244) |
OPEN |
0 inv |
2.5 |
1.5 |
29d ago |
|
396ead69 |
Runs of $>c_1\log x$ consecutive primes with all gaps $>c_2$: must they always exist? (Erdős #238) |
OPEN |
0 inv |
2.5 |
1.5 |
29d ago |
|
7a1c8d11 |
Is the number of representations $n=p+2^k$ always $o(\log n)$? (Erdős #236) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
03868985 |
Do normalized prime gaps have a continuous limiting distribution function? (Erdős #234) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
5fef66f5 |
Prove $\sum_{n\le N}(p_{n+1}-p_n)^2\ll N(\log N)^2$ for prime gaps (Erdős #233) |
OPEN |
0 inv |
3.5 |
1.5 |
29d ago |
|
2b504461 |
Are there infinitely many cluster primes? (Erdős #17) |
ACTIVE |
1 inv |
3.0 |
3.5 |
23d ago |
|
04882410 |
Does the alternating prime series $\sum(-1)^n n/p_n$ converge? (Erdős #15) |
OPEN |
0 inv |
2.5 |
1.5 |
29d ago |
|
fa409647 |
Limit points of normalized prime gaps: is $S=[0,\infty]$ for $(p_{n+1}-p_n)/\log n$? (Erdős #5) |
OPEN |
0 inv |
3.5 |
1.5 |
29d ago |
|
26339f6f |
Coprime sets in $[1,n)$: is $\sum_{a\in A}1/(n-a)\leq\sum_{p<n}1/p+O(1)$? (Erdős #1210) |
OPEN |
0 inv |
2.5 |
2.5 |
29d ago |
|
971b359f |
Diameter of admissible k-tuples: is $A(k)\sim k\log k$? (and estimate the mean $B(k)$) (Erdős #1204) |
OPEN |
0 inv |
3.5 |
2.0 |
29d ago |
|
37310009 |
Is every large integer a sum of at most $r+1$ many $r$-powerful numbers? (Erdős #1107) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
49656b48 |
Density of sums of three $k$-th powers: is $f_{k,3}(x)\gg x^{3/k}$? (Erdős #325) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
bf9e3bb8 |
A polynomial whose pairwise sums are all distinct (a polynomial Sidon set): does one exist? (Erdős #324) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
120a987f |
Density of sums of $k$-th powers: is $f_{k,k}(x)\gg x^{1-\epsilon}$ and $f_{k,m}(x)\gg x^{m/k}$? (Erdős #323) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
93c587af |
Representations as sums of $k$ many $k$-th powers: can the count exceed $n^c$ infinitely often? (Erdős #322) |
OPEN |
0 inv |
3.5 |
2.0 |
29d ago |
|
4b71a256 |
Prove the weighted shift-maximum $F(n)=\max_k\omega(n+k)\log\log k/\log k$ diverges (Erdős #1203) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
5603169c |
Least prime missing from a run of $\log n$ consecutive integers: below $(1-c)(\log n)^2$? (Erdős #1181) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
c7de7120 |
Are the $3$-smooth numbers $\{2^m3^n\}$ an essential component? (Erdős #1146) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
212bf571 |
Additive functions that rarely decrease at $n\mapsto n+1$: must they be $c\log n$? (Erdős #1122) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
f68cbd7e |
Largest subset of $\{1,\ldots,N\}$ whose pairwise sums are all squarefree (Erdős #1109) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
696cae75 |
Number of distinct primes dividing the product of the first $n$ partition numbers (Erdős #1106) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
914bd9a4 |
Growth rate of an infinite sequence whose pairwise sums are all squarefree (Erdős #1103) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
5e962925 |
Densities of EHS numbers and Pillai primes: do the counting ratios converge, and to what? (Erdős #1074) |
OPEN |
0 inv |
2.5 |
3.5 |
29d ago |
|
46dadb9f |
Count composite $u$ with $n!+1\equiv0\pmod u$ for some $n$: is $A(x)\leq x^{o(1)}$? (Erdős #1073) |
OPEN |
0 inv |
2.5 |
2.5 |
29d ago |
|
e7091b5b |
Least $n$ with $n!+1\equiv0\pmod p$: is $f(p)=p-1$ infinitely often, and $f(p)=o(p)$ a.e.? (Erdős #1072) |
OPEN |
0 inv |
2.5 |
3.5 |
29d ago |
|
f14bcb58 |
Are there infinitely many primes $p=2^kq+1$ (or $2^k3^\ell q+1$) with $q$ prime? (Erdős #1065) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
ce4d30fc |
Estimate $n_k$, least $n\geq 2k$ with $n-i\mid\binom{n}{k}$ for all but one $i<k$ (Erdős #1063) |
OPEN |
0 inv |
2.5 |
3.5 |
29d ago |
|
dc5ca039 |
Largest $A\subseteq[n]$ with no element dividing two others: is $\lim f(n)/n$ irrational? (Erdős #1062) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
e725baa9 |
Bound the multiplicity of $k\sigma(k)=n$: is the number of solutions $n^{o(1/\log\log n)}$? (Erdős #1060) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
6b17bfd5 |
Carmichael numbers: is the count $C(x)=x^{1-o(1)}$? (Erdős #1057) |
OPEN |
0 inv |
3.5 |
2.0 |
29d ago |
|
47340079 |
Consecutive integer blocks each with product $\equiv 1 \pmod p$: possible for every $k$? (Erdős #1056) |
OPEN |
0 inv |
2.0 |
4.0 |
29d ago |
|
d7330f1b |
Multiply perfect numbers: must the multiplier satisfy $k=o(\log\log n)$? (Erdős #1053) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
e6a5cff0 |
Are there only finitely many unitary perfect numbers? (Erdős #1052) |
OPEN |
0 inv |
2.0 |
2.0 |
29d ago |
|
277a09f2 |
Order of the longest similarly-ordered run of Farey fractions: is $f(n)\sim cn$? (Erdős #1005) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
902407ee |
A run of $(\log x)^c$ consecutive integers with all distinct totient values? (Erdős #1004) |
OPEN |
0 inv |
2.0 |
2.5 |
29d ago |
|
ed4d7f45 |
Are there infinitely many $n$ with $\phi(n)=\phi(n+1)$? (Erdős #1003) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
e8343875 |
A prime primitive root below every prime: does one always exist? (Erdős #985) |
OPEN |
0 inv |
2.5 |
3.5 |
29d ago |
|
80a77976 |
Estimate $f(k,n)$: primes needed to over-cover a $k$-subset of $\{1,\ldots,n\}$ (Erdős #983) |
OPEN |
0 inv |
2.0 |
2.0 |
29d ago |
|
633a2336 |
Unbounded representation counts as sums of $k$ prime $k$-th powers: is $\limsup f_k(n)=\infty$? (Erdős #979) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
181ca648 |
Are there infinitely many $n$ with $n^4+2$ squarefree? Power-free values of polynomials (Erdős #978) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
e0dd0d29 |
Greatest prime factor of $\prod_{m\le n}f(m)$: is it $\gg n^{1+c}$ for irreducible $f$? (Erdős #976) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
0b273a83 |
Infinitely many primes $p$ with $\lfloor p\alpha\rfloor$ also prime, for irrational $\alpha>1$? (Erdős #972) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
0587beee |
Are there $\gg\phi(d)$ residues $a$ with least prime $p(a,d)>(1+c)\phi(d)\log d$? (Erdős #971) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
e034b1d4 |
Order of magnitude of Jacobsthal's function $h(k)$: is $h(k)\ll k^2$? (Erdős #970) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
16bd50a7 |
Order of magnitude of the error term $E(x)$ in the count of squarefree integers (Erdős #969) |
OPEN |
0 inv |
3.5 |
2.0 |
29d ago |
|
a5f9fd41 |
Does {n : p_n/n < p_{n+1}/(n+1)} have positive density? (Erdős #968) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
96ee4052 |
Largest guaranteed dissociated subset f(n): is f(n) ≥ ⌊log₂ n⌋? (Erdős #963) |
ACTIVE |
2 inv |
3.0 |
2.0 |
15d ago |
|
5a9a3c15 |
Growth of k(n): runs of integers with a large prime factor > k (Erdős #962) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
abea354d |
Erdős–Granville–Pomerance–Spiro: does density 0 pull back to density 0 under s(n)? (Erdős #955) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
11aa123d |
Error term for Rosen's greedy B_2-type sequence: is R(x)=x+O(x^{1/4+o(1)})? (Erdős #954) |
OPEN |
0 inv |
2.0 |
3.5 |
29d ago |
|
35f2b18b |
Gaussian moat: is there an infinite bounded-step walk on Gaussian primes? (Erdős #952) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
c479ce46 |
Do Beurling generalised primes satisfy #{a_i ≤ x} ≤ π(x)? (Erdős #951) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
1168e89a |
Is the two-powerful-number representation function n^{o(1)}? (Erdős #943) |
OPEN |
0 inv |
2.0 |
2.5 |
29d ago |
|
a9ed455c |
Estimate $S(k)$, the least $x$ forcing dense $k$-runs each divisible by a prime $\leq x$ (Erdős #929) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
85b24440 |
Does the density of $n$ with $P(n)<n^\alpha$ and $P(n+1)<(n+1)^\beta$ exist? (Erdős #928) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
928bd37d |
Infinitely many $n$ with all exponents in the factorisation of $n(n+1)$ distinct? (Erdős #913) |
OPEN |
0 inv |
2.5 |
2.5 |
29d ago |
|
d3feaa35 |
Must every length-$p_1\cdots p_k$ interval contain an integer with $>k$ prime factors? (Erdős #891) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
136f5ccb |
Erdős–Selfridge: does the peak count of large 'new' prime factors $v_0(n)$ tend to infinity? (Erdős #889) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
9174225d |
Maximal sum of a pairwise-coprime subset of $\{1,\ldots,n\}$: is $G(n)>H(n)-n^{1+o(1)}$? (Erdős #879) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
ac9c766e |
Extremal order and coincidence of the prime-power functions $f(n)$ and $F(n)$ (Erdős #878) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
d81452b3 |
Sliding-window LCM counts of a sequence: can $F(A,X,k)<X^\epsilon$ be forced for some $k$? (Erdős #873) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
7ce72216 |
Growth of $H(n)$, least $l$ with $\gcd(k^n\!-\!1,l^n\!-\!1)=1$ for some $k<l$: is $H(n)=3$ i.o.? (Erdős #820) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
bf25eb1c |
Second-order term of $g_3(n)$: largest $A\subseteq[n]$ with every product $<3$ times represented (Erdős #796) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
5744742c |
A near-density-1 set whose equal products of distinct elements have equally many factors (Erdős #786) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
89bcce09 |
Do the squares contain arbitrarily long quasi-progressions and arbitrarily large cubes? (Erdős #782) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
98148417 |
Is every proportionately dissociated set a finite union of dissociated sets? (Erdős #774) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
f3d8a75e |
Density and liminf of $h(n)$, least $l$ making $2^n\!-\!1,\ldots,l^n\!-\!1$ pairwise coprime (Erdős #770) |
OPEN |
0 inv |
2.5 |
2.5 |
29d ago |
|
5f9b6ec8 |
Restricted Mertens sum over primes with $n\bmod p\in(p/2,p)$: is it $\sim\tfrac12\log\log n$? (Erdős #726) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
f0166e1d |
Bound $f(n,m)$ for distinct multiples $k\mid a_k$: is $\max_m f(n,m)\le n^{1+o(1)}$? (Erdős #711) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
7ec2e726 |
Distinctness of consecutive-block lcms: is $M(n,k)\neq M(m,k)$ whenever $m\ge n+k$? (Erdős #677) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
51143567 |
Is every large integer of the form $ap^2+b$ with $p$ prime, $a\ge1$, $0\le b<p$? (Erdős #676) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
00d54a95 |
Translation property: sums of two squares, prime-restricted sets, and squarefree shift growth (Erdős #675) |
OPEN |
0 inv |
2.5 |
2.5 |
29d ago |
|
9f279e68 |
Least prime missing from a product of $k$ consecutive integers: is $q(n,k)<(1+o(1))\log n$? (Erdős #663) |
OPEN |
0 inv |
2.0 |
2.0 |
29d ago |
|
5ccf31c6 |
Estimate $h(n)$: fewest distinct ratios $a/\gcd(a,b)$ forced by an $n$-element set (Erdős #539) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
52d0e6a5 |
Best-possible upper bound for $\sum_{n\in A}1/n$ under an at-most-$r$ prime-representation cap (Erdős #538) |
OPEN |
0 inv |
2.0 |
2.0 |
29d ago |
|
d74129a9 |
Estimate $f_r(N)$: largest subset of $\{1,\ldots,N\}$ with no $r$ elements sharing one pairwise gcd (Erdős #535) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
b6667243 |
Second moment of gaps among non-multiples of a sparse set: does the limit exist? (Erdős #489) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
9fbc536c |
Graham's conjecture: for every $k\neq 1$, infinitely many $n$ with $2^n\equiv k\pmod{n}$? (Erdős #479) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
90377b0a |
Exact additive complement of a degree-$\geq 2$ polynomial image: does one exist? (Erdős #477) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
3aa15e1f |
Ulam's greedy prime sequence $q_{n+1}=$ least prime $q_n+q_i-1$: can it be infinite? (Erdős #472) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
4f9fad7b |
Two-part prime congruence cover: split $\{p\leq x\}$ so every $n<x$ is hit in both parts (Erdős #467) |
OPEN |
0 inv |
2.0 |
2.0 |
29d ago |
|
1e833fbd |
Divergence of $\sum 1/a_i$ for the Eggleton–Erdős–Selfridge coprime sequence (Erdős #460) |
OPEN |
0 inv |
2.5 |
2.5 |
29d ago |
|
964173a6 |
Smallest prime $\equiv 1\ (\mathrm{mod}\ n)$ versus smallest $m$ with $n\mid\phi(m)$ (Erdős #456) |
OPEN |
0 inv |
2.5 |
2.5 |
29d ago |
|
933a3949 |
Convex-gap prime sequences: must $q_n/n^2\to\infty$? (Erdős #455) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
708a7e90 |
Longest run in $[x,2x]$ of integers with more than $\log\log n$ distinct prime factors (Erdős #452) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
84d66419 |
Distinct-distance subsets: estimate the guaranteed size $F_d(n)$ in any $n$ points of $\mathbb{R}^d$ (Erdős #1208) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
4876423e |
Factor-difference sets: do $k$ integers always share $\geq k$ common factor differences? (Erdős #885) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
11a9e739 |
Density $d_t$ of $n$ representing $t$ as a sum of distinct divisors: is $d_t\sim c_1(\log t)^{-c_2}$? (Erdős #859) |
OPEN |
0 inv |
2.0 |
2.5 |
29d ago |
|
c9e48276 |
Four-point near-Sidon sets: the best constant $c$ forcing a Sidon subset of size $cn$ (Erdős #757) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
85a4a384 |
Intervals nearly free of integers with a divisor in $(n,2n)$: how large must $y(\epsilon,n)$ be? (Erdős #450) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
0627383b |
Practical numbers with tiny representations: is $h(m)<(\log\log m)^{O(1)}$ infinitely often? (Erdős #18) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
88bbdc31 |
Lagrange interpolation at Chebyshev nodes: realise every closed set as its limit points (Erdős #1151) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
dfd2930b |
Node sets forcing every low-degree near-interpolant to exceed a fixed bound (Erdős #1133) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
04244230 |
Largest measure of a bounded planar set with no two points an integer distance apart (Erdős #953) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
dae93785 |
Distinct distances under a no-three-concyclic-per-centre condition: at least $(1+c)n/2$? (Erdős #655) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
bacde559 |
Cochromatic gap of the random graph: is $\chi(G)-\zeta(G)\to\infty$ almost surely? (Erdős #625) |
OPEN |
0 inv |
4.0 |
1.0 |
29d ago |
|
9e1b354e |
Is the largest disc inside $\{|f|<1\}$ of radius $\gg 1/n$ for roots in the unit disc? (Erdős #1039) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
9de66620 |
An entire function whose every derivative-subsequence has dense zero set: does one exist? (Erdős #906) |
OPEN |
0 inv |
2.0 |
1.0 |
29d ago |
|
5724ea9e |
Can Lagrange interpolation converge while the Lebesgue function diverges? (Erdős #671) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
a5d64348 |
Bound the length of a path along which an entire function outgrows every power $z^n$ (Erdős #514) |
OPEN |
0 inv |
2.0 |
2.0 |
29d ago |
|
ba2d107e |
Does a minimal order-2 additive basis with $a_k\sim ck^2$ exist? (Erdős #326) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
ab8cc421 |
Estimate $h(n)$, the powerful integers in $[n^2,(n+1)^2)$: is it $(\log n)^{c+o(1)}$? (Erdős #942) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
e706f515 |
Integers that are no sum of $r$ many $r$-powerful numbers: infinitely many, sumset density 0? (Erdős #940) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
063b8c26 |
Can a sum of $r-2$ coprime $r$-powerful numbers be $r$-powerful (open case $r=4$)? (Erdős #939) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
e7603de8 |
Finitely many 3-term arithmetic progressions among consecutive powerful numbers? (Erdős #938) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
17de0d4c |
Are $2^n\pm1$ and $n!\pm1$ powerful for only finitely many $n$? (Erdős #936) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
55e3d2b6 |
Is the $\{2,3\}$-part of $n(n+1)$ infinitely often much larger than $n\log n$? (Erdős #933) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
c9854261 |
Two integers between consecutive primes with all prime factors below the gap, infinitely often (Erdős #932) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
901abe42 |
Products of consecutive integers over disjoint long intervals: never a perfect power? (Erdős #930) |
OPEN |
0 inv |
3.5 |
1.5 |
29d ago |
|
4b854247 |
Gaps between totatives of a primorial: which even numbers occur, and how often? (Erdős #854) |
OPEN |
0 inv |
2.5 |
4.0 |
29d ago |
|
62217b5c |
No term a sum of consecutive earlier terms: must $\limsup a_n/n=\infty$? (Erdős #839) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
89323dcf |
Are there infinitely many amicable pairs, and is $A(x)>x^{1-o(1)}$? (Erdős #830) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
1af18651 |
Sums of two cubes: is the representation count $1_A*1_A(n)\ll(\log n)^{O(1)}$? (Erdős #829) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
6fba68f8 |
Graham's conjecture: infinitely many $n$ with $\phi(n)\mid n+a$ for every $a$? (Erdős #828) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
64350cb5 |
Infinitely many $n$ with $\tau(n+k)\ll k$ for all $k\geq 1$? (Erdős #826) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
c689c7f0 |
Count coprime pairs with equal sum-of-divisors: is $h(x)>x^{2-o(1)}$? (Erdős #824) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
c98d9a74 |
Erdős–Pomerance: asymptotics of the window $(n,n+f(n))$ holding distinct multiples of $1,\ldots,n$ (Erdős #710) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
127598a0 |
Erdős–Surányi distinct multiples in a window: bound $f(n)$ between $\log n/\log\log n$ and $\sqrt n$ (Erdős #709) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
a29f5ba4 |
Erdős–Surányi product divisibility: is $g(n)\leq(2+o(1))n$? (Erdős #708) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
4c95a5df |
Growth of prime chains $p_{i+1}\equiv 1\pmod{p_i}$: is $\lim_k p_k^{1/k}=\infty$? (Erdős #695) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
553bcdfc |
Characterise the Behrend sequences: which $A$ make the set of multiples $M_A$ have density 1? (Erdős #691) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
44cee90e |
Covering $[1,n]$ by residues of only the large primes: estimate $\epsilon_n$; is $\epsilon_n=o(1)$? (Erdős #688) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
8f16e35a |
Estimate the Jacobsthal-type covering maximum $Y(x)$: is $Y(x)=o(x^2)$? (Erdős #687) |
OPEN |
0 inv |
4.0 |
2.0 |
29d ago |
|
20944fcf |
Estimate $n_k$: least $n>2k$ with $(n-1)(n-2)\cdots(n-k)$ having no prime factor in $(k,2k)$ (Erdős #451) |
ACTIVE |
2 inv |
3.0 |
3.0 |
28d ago |
|
e3ce6737 |
For $c>1/2$ and large $p$, does every interval $(n,n+p^c)$ contain $a,b$ with $ab\equiv1\pmod p$? (Erdős #445) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
c9f313ff |
Is $\Lambda(k,3)$ finite for all odd $k$, and how fast do $\Lambda(k,2),\Lambda(k,3)$ grow? (Erdős #436) |
ACTIVE |
2 inv |
3.0 |
3.5 |
23d ago |
|
63c2f652 |
How dense can the sumset $A+B$ be if all its elements are pairwise coprime? (Erdős #432) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
2bd31447 |
For large $n$, must the greedy $[1,n)$ sequence with all prime factors $>n-a$ include a composite? (Erdős #430) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
15ab61af |
Does the sequence built from $2,3$ by adjoining all $a_ia_j-1$ have positive density? (Erdős #424) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
9f8d9815 |
Growth of the greedy sequence whose terms are the least new sum of $\ge 2$ consecutive earlier terms (Erdős #423) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
3ec60c1b |
Does the Hofstadter Q-sequence $f(n)=f(n-f(n-1))+f(n-f(n-2))$ miss infinitely many integers? (Erdős #422) |
OPEN |
0 inv |
2.5 |
2.5 |
36d ago |
|
6a8e8519 |
Density and growth of $\tau((n+f(n))!)/\tau(n!)$, ratios of divisor-counts of nearby factorials (Erdős #420) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
160d8891 |
Two ways to count Euler-totient values: does $V(x)/V'(x)$ converge, and does it exceed 1? (Erdős #417) |
OPEN |
0 inv |
2.5 |
3.0 |
36d ago |
|
9e60feb0 |
Distribution of Euler-totient values: does $V(2x)/V(x)\to 2$, with an asymptotic for $V(x)$? (Erdős #416) |
OPEN |
0 inv |
3.5 |
2.5 |
36d ago |
|
16efa709 |
Summatory growth of $t_k(n)$, the least start making $n$ divide a run of $k$ consecutive integers (Erdős #394) |
OPEN |
0 inv |
2.5 |
2.0 |
36d ago |
|
03500b7e |
Does every n admit a k with the product of k consecutive integers from n dividing the next k? (Erdős #389) |
OPEN |
0 inv |
2.5 |
3.5 |
36d ago |
|
63ce256d |
Are there only finitely many equal products of two disjoint blocks of 4+ consecutive integers? (Erdős #388) |
ACTIVE |
1 inv |
2.5 |
3.0 |
15d ago |
|
5cc91e89 |
Is $\max_{m<n}(m+p(m))>n$ eventually and does the excess diverge? (Erdős #385) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
1dc57ca1 |
Infinitely many primes $p$ with top prime factor of $\prod_{0\le i\le k}(p^2+i)$ equal to $p$? (Erdős #383) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
a9c6ac49 |
Runs of consecutive integers whose product's top prime is squared: can $v-u$ be unbounded? (Erdős #382) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
316f41fc |
How fast does $|D_k\cap[1,n]|$ grow for the factorial-product-square index $F(m)=k$? (Erdős #374) |
OPEN |
0 inv |
3.0 |
4.0 |
36d ago |
|
528b3173 |
Does $\{n : P(n)<P(n+1)\}$ have natural density exactly $1/2$? (Erdős #371) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
df853a01 |
Growth of $F(n)$, the largest prime factor of $n(n+1)$: how small can it be? (Erdős #368) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
96f33311 |
Bound the product of the 2-full parts of $k$ consecutive integers: is it $n^{2+o(1)}$? (Erdős #367) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
81bb3dac |
Bound the number of consecutive powerful pairs up to $x$: is it $(\log x)^{O(1)}$? (Erdős #365) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
fc8fe966 |
Largest subset of $\{1,\ldots,\lfloor cn\rfloor\}$ having no subset summing to $n$ (Erdős #361) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
f5fdaa68 |
Density and growth of MacMahon's prime numbers of measurement (segmented numbers) (Erdős #359) |
OPEN |
0 inv |
2.5 |
3.0 |
36d ago |
|
5e28fa54 |
Growth of $f(n)$: largest increasing set in $[n]$ with all consecutive-block sums distinct (Erdős #357) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
7a5c5cc0 |
Completeness of $\{\lfloor 2^k\alpha\rfloor\}\cup\{\lfloor 2^k\beta\rfloor\}$ for irrational $\alpha/\beta$ (Erdős #354) |
OPEN |
0 inv |
2.5 |
2.0 |
36d ago |
|
0bf09014 |
Ulam numbers: twin pairs, eventual gap-periodicity, and zero density (Erdős #342) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
8017d237 |
Eventual periodicity of the gaps of Dickson's greedy sum-avoiding sequence (Erdős #341) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
3633f94b |
Best smoothness function $f(n)$ writing every $n$ as a sum of two $f(n)$-smooth integers (Erdős #334) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
c29b53eb |
Sufficient conditions for the infinitely-recurring difference set $D(A)$ to have bounded gaps (Erdős #332) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
f2bf4f3a |
How dense can an infinite Sidon set be along N^{1/2}? (Erdős #329) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
cf4e1d54 |
Is n/2^n always a finite sum of distinct terms a/2^a? (Erdős #261) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
33e9b8e2 |
A density and equidistribution condition forcing subset-sum completeness (Erdős #254) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
994de563 |
Must a near-squaring integer sequence with rational reciprocal sum be Sylvester's sequence? (Erdős #243) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
7e911005 |
How large can gaps between consecutive squarefree numbers be? (Erdős #208) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
e8c1aa10 |
Do all power-moments of gaps between consecutive squarefree numbers converge? (Erdős #145) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
2f4779cc |
Can a product of k ≥ 3 consecutive integers ever be powerful? (Erdős #137) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
1443d057 |
Estimate the maximum size of a non-dividing subset of {1,...,N} (Erdős #131) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
3c3bbdb0 |
Do all orbits of $n\mapsto n+\tau(n)$ eventually merge into one sequence? (Erdős #414) |
OPEN |
0 inv |
2.0 |
3.5 |
36d ago |
|
94eef7cb |
Fewest primes dividing all pairwise sums of an $n$-set: is $f(n)/\log n\to\infty$? (Erdős #126) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
ce672d54 |
Is $\{a^k b^l c^m\}$ d-complete for every pairwise-coprime $a,b,c$? (Erdős #123) |
ACTIVE |
1 inv |
3.0 |
4.0 |
23d ago |
|
f7589ebe |
For which arithmetic functions $f$ do the values $n+f(n)$ cluster into short intervals? (Erdős #122) |
OPEN |
0 inv |
2.5 |
1.5 |
36d ago |
|
4965cda5 |
An infinite set of totient values whose smallest preimages grow superlinearly? (Erdős #51) |
ACTIVE |
1 inv |
2.0 |
2.5 |
15d ago |
|
d60a3921 |
Is the distribution function of $\varphi(n)/n$ nowhere of positive derivative? (Erdős #50) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
8244bfcd |
Must the surviving set of an arbitrary congruence sieve have a logarithmic density? (Erdős #25) |
OPEN |
0 inv |
2.0 |
1.0 |
36d ago |
|
a883df83 |
Can a set where no member divides the sum of two larger members have divergent reciprocal sum? (Erdős #12) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
a18606bb |
Do all iterated-$\sigma$ orbits eventually merge: $\sigma_i(m)=\sigma_j(n)$ for some $i,j$? (Erdős #412) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
a9009d31 |
Eventual-doubling of the $n+\phi(n)$ iteration: which $n,r$ give $g_{k+r}(n)=2g_k(n)$? (Erdős #411) |
ACTIVE |
1 inv |
2.5 |
3.5 |
15d ago |
|
f753f680 |
Does iterated $\sigma$ grow super-exponentially: $\lim_k \sigma_k(n)^{1/k}=\infty$ for all $n\ge2$? (Erdős #410) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
d088c814 |
Iterating $n\mapsto\phi(n)+1$ to a prime: iteration count, fibers, and densities (Erdős #409) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
956ee156 |
Distribution of $f(n)=\min\{k:\phi_k(n)=1\}$, the totient iteration length (Erdős #408) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
7b875624 |
Is $\sum_n \sigma_k(n)/n!$ irrational for every $k\ge1$? (Erdős #252) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
97454cf4 |
Is $\sum_n p_n/2^n$ irrational, where $p_n$ is the $n$th prime? (Erdős #251) |
OPEN |
0 inv |
2.0 |
1.0 |
36d ago |
|
4795295d |
Is $\sum_n \phi(n)/2^n$ irrational, where $\phi$ is Euler's totient? (Erdős #249) |
OPEN |
0 inv |
2.0 |
1.0 |
36d ago |
|
3d272410 |
Chowla's conjecture: is $\sum 1/(t^n-1)$ irrational for every rational $t>1$? (Erdős #1049) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
adb97de4 |
Is the reciprocal sum of running LCMs of $P$-smooth numbers irrational? (Erdős #269) |
OPEN |
0 inv |
2.0 |
1.5 |
36d ago |
|
595699ec |
Irrationality of $\sum 1/F_{n_k}$ for lacunary Fibonacci subsequences with ratio $c\in(1,2)$ (Erdős #267) |
OPEN |
0 inv |
2.5 |
1.5 |
36d ago |
|
da95fd3a |
How fast can $a_n$ grow if $\sum 1/a_n$ and $\sum 1/(a_n-1)$ are both rational? (Erdős #265) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
5115409f |
Is $n!$ a perturbation-robust irrationality sequence? ($2^n$ is now known to fail) (Erdős #264) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
832c26a4 |
Irrationality sequences: is $2^{2^n}$ one, and must every such sequence satisfy $a_n^{1/n}\to\infty$? (Erdős #263) |
OPEN |
0 inv |
2.5 |
1.0 |
36d ago |
|
75b6cbf8 |
Transcendence of the binary sum $\sum 1/2^{a_n}$ when $\limsup a_n/n=\infty$ (Erdős #247) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
db9299d6 |
Is $\sum_{n\ge 2} 1/(n!-1)$ irrational? (Erdős #68) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
60732969 |
Asymptotic formula for the number of subgroups of the symmetric group $S_n$ (Erdős #1162) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
1260e6dc |
Do powers of 2 maximise the group-count: is $g(n)\le g(2^m)$ for all $n\le 2^m$? (Erdős #1160) |
OPEN |
0 inv |
2.5 |
1.5 |
36d ago |
|
1cd0b40d |
Is the Turán number of $K_t(r)$ (complete $t$-partite $t$-uniform) at least $n^{t-r^{1-t}-o(1)}$? (Erdős #1158) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
13a60f2d |
Determine the Brown–Erdős–Sós Turán number: max edges with no $k$ vertices spanning $s$ edges (Erdős #1157) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
fc0a8cf0 |
Do dense $r$-uniform hypergraphs contain growing subgraphs of density above $r^{-r}$? (Erdős #1075) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
7611880a |
Is $\sum_{n\in A}1/(2^n-1)$ irrational for every infinite set $A\subseteq\mathbb{N}$? (Erdős #257) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
7e1de0cf |
Minimum Turán number over $k$-vertex, $l$-edge graphs: estimate $f(n;k,l)$ and its monotonicity (Erdős #766) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
08723f0e |
Tightness of the Kővári–Sós–Turán bound: is $\mathrm{ex}(n;K_{r,r})\gg n^{2-1/r}$? (Erdős #714) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
dac5e12e |
Do bipartite Turán numbers have the form $c\,n^\alpha$ with rational $\alpha$? (Erdős #713) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
138dfa37 |
Does a dense $K_{2,2,2}$-free graph force a linear-size independent set? (Erdős #579) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
c074a458 |
Turán number of the hypercube $Q_k$: determine $\mathrm{ex}(n;Q_k)$ (is $\mathrm{ex}(n;Q_3)\asymp n^{8/5}$?) (Erdős #576) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
f8924fc6 |
Is a family's Turán number governed by one bipartite member? (Erdős #575) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
5c56e2dd |
Maximum edges in a girth-5 graph: is $\mathrm{ex}(n;\{C_3,C_4\})\sim(n/2)^{3/2}$? (Erdős #573) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
3b75f8c5 |
Covering near-abelian groups by abelian subgroups: estimate $h(n)$ (Erdős #117) |
OPEN |
0 inv |
2.5 |
1.0 |
36d ago |
|
7cf78523 |
Which limit ordinals $\alpha$ force every graph on $\alpha$ to have an infinite path or an independent set of type $\alpha$? (Erdős #601) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
2e17d326 |
Does $\omega_1^2\to(\omega_1\omega,G)^2$ hold for every $K_4$-free, $K_{\aleph_0,\aleph_0}$-free graph $G$? (Erdős #597) |
OPEN |
0 inv |
2.5 |
1.0 |
36d ago |
|
c7c05a58 |
Characterize the graph pairs $(G_1,G_2)$ with a finite-colour vs $\aleph_0$-colour Ramsey gap (Erdős #596) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
0ab515f5 |
An infinite $K_4$-free graph that is not a countable union of triangle-free graphs: does one exist? (Erdős #595) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
9aae1126 |
Even-cycle Turán lower bound: is $\mathrm{ex}(n;C_{2k})\gg n^{1+1/k}$ for every $k\geq 3$? (Erdős #572) |
OPEN |
0 inv |
4.0 |
1.5 |
36d ago |
|
44566412 |
Rational Turán exponents: is every rational $\alpha\in[1,2)$ the exponent of $\mathrm{ex}(n;G)$ for some bipartite $G$? (Erdős #571) |
OPEN |
0 inv |
4.0 |
1.5 |
36d ago |
|
3c43528e |
For a finite forbidden family $\mathcal{F}$, does some $G\in\mathcal{F}$ have $\mathrm{ex}(n;G)\asymp\mathrm{ex}(n;\mathcal{F})$? (Erdős #180) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
d16343c2 |
Degenerate Turán conjecture: does $r$-degenerate bipartite $H$ force $\mathrm{ex}(n;H)\ll n^{2-1/r}$? (Erdős #146) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
5386126d |
Maximum edges keeping $R(K_3,G)=2n-1$: estimate $f(n)$ and $F(n)$ (Erdős #1182) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
70d10f0b |
Near-diagonal Ramsey ratio: is $R(k+1,k)/R(k,k)\geq 1+c$? (Erdős #1030) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
fc281228 |
Does $R(k)/(k\,2^{k/2})\to\infty$? Beat the probabilistic diagonal Ramsey lower bound (Erdős #1029) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
191bda90 |
Size Ramsey number of dense graphs: is $\hat R(G)$ superlinear in the edge count? (Erdős #911) |
OPEN |
0 inv |
2.0 |
1.0 |
36d ago |
|
41262e66 |
Growth of consecutive diagonal Ramsey numbers: is $R(n+1)/R(n)\geq 1+c$? (Erdős #812) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
048ca1fd |
Balanced $e(G)$-colourings of $K_n$: which graphs $G$ are forced to appear rainbow? (Erdős #811) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
abb27b3a |
Can a graph with $\epsilon n^2$ edges be $n$-coloured so every $C_4$ is rainbow? (Erdős #810) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
6ce28459 |
The partition relation $\mathfrak{c}\to(\beta,n)^3_2$ for countable $\beta$ and finite $n$ (Erdős #70) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
8643a05d |
Symmetric anti-Ramsey number for odd cycles: settle the last open case $C_7$ (Erdős #809) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
248b1542 |
Is the local-density Ramsey exponent $c(p,q)$ strictly increasing in $q$? (Erdős #667) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
12f78549 |
Estimate $f(n)$: the shortest monochromatic odd cycle forced in $n$-colourings of $K_{2^n+1}$ (Erdős #609) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
589c2ced |
Do linear tree-Ramsey and quadratic clique-Ramsey together force Ramsey size-linearity? (Erdős #568) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
8243922c |
Ramsey size-linearity of $Q_3$, $K_{3,3}$, and the subdivided $K_4$: is $R(G,H)\ll m$? (Erdős #567) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
8a48a84f |
Is every graph whose $k$-vertex subgraphs have at most $2k-3$ edges Ramsey size-linear? (Erdős #566) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
1928225e |
Size Ramsey number of star forests: prove $\hat{R}(F_1,F_2)=\sum_k\max\{n_i+m_j-1\}$ (Erdős #561) |
OPEN |
0 inv |
2.5 |
2.5 |
36d ago |
|
c45beb19 |
Determine the size Ramsey number $\hat{R}(K_{n,n})$ of the complete bipartite graph (Erdős #560) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
e346503e |
Determine the multicolour Ramsey number $R_k(K_{s,t})$ of complete bipartite graphs (Erdős #558) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
5df8b87b |
Do multicolour Ramsey numbers of trees grow linearly: is $R_k(T)\leq kn+O(1)$? (Erdős #557) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
fbd34fd1 |
Determine the multicolour Ramsey number $R_k(C_{2n})$ of even cycles (Erdős #555) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
5c5fd7bb |
Multicolour Ramsey of odd cycles negligible vs triangles: $R_k(C_{2n+1})/R_k(K_3)\to0$ (Erdős #554) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
de1bde1f |
Determine the Ramsey number $R(C_4,S_n)$ of a 4-cycle versus a star (Erdős #552) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
e89ddd72 |
Is the Ramsey number $R(G)$ over $m$-edge graphs maximised by the 'almost complete' graph? (Erdős #545) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
f835e3d0 |
Do consecutive Ramsey gaps $R(3,k+1)-R(3,k)$ tend to infinity, and are they $o(k)$? (Erdős #544) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
afcfec75 |
Determine $\lim_k R(3;k)^{1/k}$ for the multicolour triangle Ramsey number (Erdős #183) |
OPEN |
0 inv |
4.5 |
2.0 |
36d ago |
|
92499258 |
Prove $R(Q_n)\ll 2^n$: is the Ramsey number of the hypercube linear in its vertex count? (Erdős #181) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
e2ffee3b |
Give an asymptotic formula for $R(3,k)$: pin the constant in $k^2/\log k$ (Erdős #165) |
OPEN |
0 inv |
4.5 |
1.5 |
36d ago |
|
7d55c64a |
Prove a power saving $R(C_4,K_n)\ll n^{2-c}$ for the 4-cycle vs clique Ramsey number (Erdős #159) |
OPEN |
0 inv |
3.5 |
1.0 |
36d ago |
|
50ff2c2a |
Determine the digraph Ramsey function $k(n,m)$: independent set vs transitive tournament (Erdős #112) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
a8284804 |
Independence number of planar minimum-distance-1 point sets: estimate $g(n)$ and $\lim g(n)/n$ (Erdős #1066) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
f862d502 |
Coprime graph of a dense subset of $[n]$: does the extremal threshold force all short odd cycles? (Erdős #883) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
75327590 |
Turán density of the complete $r$-graph $K_k^r$ for every fixed $k>r>2$ (Erdős #712) |
OPEN |
0 inv |
4.5 |
2.5 |
36d ago |
|
68a826c5 |
Turán density of the tetrahedron $K_4^3$: evaluate $\lim \mathrm{ex}_3(n,K_4^3)/\binom{n}{3}$ (Erdős #500) |
OPEN |
0 inv |
4.5 |
2.5 |
36d ago |
|
5161b7cf |
Does chromatic number $k$ force the Ramsey number $R(G)$ close to $R(k)$? (Erdős #87) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
67078ae2 |
Book size forced in dense graphs covered by triangles: estimate $f_c(n)$, is it $\gg\log n$? (Erdős #80) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
4a822fce |
Constructive exponential lower bound for Ramsey numbers: explicit graphs forcing $R(k)>C^k$ (Erdős #78) |
OPEN |
0 inv |
3.5 |
1.5 |
36d ago |
|
ebb7504d |
Determine the diagonal Ramsey growth constant $\lim_{k\to\infty} R(k)^{1/k}$ (Erdős #77) |
OPEN |
0 inv |
4.5 |
1.0 |
36d ago |
|
fcde6c7d |
Brown–Erdős–Sós conjecture: is the $o(n^2)$ threshold $d_r(e)=(r-2)e+3$? (Erdős #1178) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
ca38206a |
The random triangle-removal process: does the surviving edge count $f(n)$ scale as $n^{3/2}$? (Erdős #1155) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
86774d3d |
Determine $A_3$, the set of jump densities for $3$-uniform hypergraphs (Erdős–Simonovits) (Erdős #837) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
f544b1e3 |
Erdős–Sauer conjecture: decompose every $r$-uniform hypergraph into few cliques and single edges (Erdős #719) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
d3e14bd7 |
Extremal edge count forcing two disjoint edge-pairs with equal union in a $t$-uniform hypergraph (Erdős #643) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
45be4a28 |
Does the $3$-uniform hypergraph Ramsey number satisfy $R_3(n)\geq 2^{2^{cn}}$? (Erdős #564) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
40e838be |
Sharp $c_\alpha\log n$ asymptotic for the two-colour density-$\alpha$ subgraph threshold (Erdős #563) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
9330cf51 |
Hypergraph Ramsey tower height: does $R_r(n)$ grow like a height-$(r-1)$ tower in $n$? (Erdős #562) |
OPEN |
0 inv |
3.5 |
1.0 |
36d ago |
|
87592d1b |
Must large chromatic number with no K_t force two anticomplete c-chromatic subgraphs? (Erdős #1111) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
b5105156 |
A minimum-degree threshold on 2^n vertices forcing the n-cube Q_n (Erdős #1035) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
6c1038e9 |
Estimate h(n): largest guaranteed triangle degree-sum above the Turán threshold (Erdős #1033) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
cd278e5a |
Estimate f(n,k), the clique partition number for graphs with more than n²/4 edges (Erdős #1017) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
8f2f325f |
Determine h_3(k): fewest vertices in a triangle-free graph of chromatic number k (Erdős #1013) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
62208b79 |
Determine f_r(n): fewest edges forcing a triangle in an n-vertex graph of chromatic number ≥ r (Erdős #1011) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
7484ed39 |
Estimate h_t(d): fewest edges forcing two edges at distance ≥ t in a max-degree-d graph (Erdős #934) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
e36d5e7b |
Estimate f(n): fewest vertices in a tournament where every n vertices have a common dominator (Erdős #902) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
ae16c550 |
Erdős–Hajnal: clique size forced when every 7 vertices span a triangle — estimate $h(n)$ (Erdős #813) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
f4b54a16 |
Erdős–Hajnal: smallest $g(n)$ so every $g(n)$-subset has a $\log n$ clique and $\log n$ independent set (Erdős #805) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
067f65f8 |
Independence number of $K_r$-free graphs: is the AEKS $\frac{\log t}{t}n$ bound true for all $r$? (Erdős #802) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
2325b8ed |
Erdős's Alice–Bob clique game on $K_n$: does Bob have a winning strategy for all $n\geq 3$? (Erdős #778) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
0e1e781a |
Erdős–Rogers problem: largest triangle-free induced subgraph forced in a $K_4$-free graph (Erdős #620) |
OPEN |
0 inv |
4.0 |
1.5 |
36d ago |
|
d6b3ed12 |
Pin down $t(r)$: transversal number forced by a local $\tau\leq 1$ condition on $r$-uniform hypergraphs (Erdős #616) |
ACTIVE |
3 inv |
3.0 |
2.0 |
15d ago |
|
40f3f739 |
Determine $f(n,k)$: fewest edges forcing degree $\geq k$ in every $(k+2)$-vertex induced subgraph (Erdős #614) |
OPEN |
0 inv |
2.0 |
3.5 |
36d ago |
|
c6a367a7 |
Diameter of $K_{k+1}$-free graphs with minimum degree $d$: is it at most $(3-2/k)n/d$? (Erdős #612) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
eb2b00ba |
Sublinear clique transversals under a large-clique hypothesis: is $\tau(G)=o_c(n)$? (Erdős #611) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
54592ee7 |
Edges forcing an $r$-triangle edge: are the thresholds $e(n,r)$ asymptotically flat in $r$? (Erdős #600) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
c6a62326 |
Clique transversal vs. independence: is $\tau(G)\le n-H(n)$ for all graphs? (Erdős #151) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
9d8169b1 |
Strong chromatic index conjecture: is $\mathrm{sq}(G)\le\tfrac54\Delta^2$ for every graph? (Erdős #149) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
6907909b |
Turán density of $C_4$ in the hypercube: does $(1/2+o(1))n2^{n-1}$ edges force a $C_4$? (Erdős #86) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
c7e81a65 |
Is $f(n)$ — the min-degree threshold forcing a $C_4$ — eventually monotonic? (Erdős #85) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
781d464a |
Force a large regular induced subgraph: does $F(n)/\log n\to\infty$? (Erdős #82) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
ac3b55c4 |
Partition the edges of a chordal graph into cliques: is $n^2/6+O(n)$ always enough? (Erdős #81) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
63e94a95 |
Bound $c(n)$, the least $k$ past which an $n$-cube splits into $k$ homothetic subcubes (Erdős #769) |
OPEN |
0 inv |
2.5 |
2.5 |
36d ago |
|
846c7138 |
Self-avoiding walk displacement: does $d_2(n)/\sqrt{n}\to\infty$ and $d_k(n)\ll\sqrt{n}$ for $k\geq3$? (Erdős #529) |
OPEN |
0 inv |
3.5 |
2.0 |
36d ago |
|
745418e0 |
Chromatic number of the plane (Hadwiger–Nelson): pin $\chi(\mathbb{R}^2)$ between 5 and 7 (Erdős #508) |
OPEN |
0 inv |
4.5 |
2.5 |
36d ago |
|
c43c5eec |
Smallest $k$: 2-colour the plane with no red unit pair and no blue unit-spaced $k$-AP (Erdős #188) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
a41287a4 |
Characterise the Ramsey finite point sets in Euclidean space (Erdős #174) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
38f9bab2 |
Monochromatic triangles under any 2-colouring of the plane: at most one exceptional shape? (Erdős #173) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
25026bec |
Common finite-chromatic subgraph of two graphs of chromatic number $\aleph_1$ (Erdős #62) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
b83472a9 |
Erdős–Hajnal conjecture: does an excluded induced $H$ force a polynomial clique or independent set? (Erdős #61) |
OPEN |
0 inv |
4.5 |
1.0 |
36d ago |
|
997fb055 |
Points in $\mathbb{R}^d$ forcing $n$ with all pairwise distances distinct: is $f_d(n)=2^{o(d)}$? (Erdős #1088) |
OPEN |
0 inv |
2.0 |
1.5 |
36d ago |
|
3d9e309e |
Estimate $h(n)$: distinct-radius circles forced through triples of $n$ planar points (Erdős #831) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
d8aac4b1 |
Determine $n_k$: fewest general-position points forcing $k$ whose triples give all-distinct circle radii (Erdős #827) |
OPEN |
0 inv |
2.0 |
1.5 |
36d ago |
|
2bffc76c |
Generalized orchard problem: determine $\lim F_k(n)/n^2$ and $\lim f_k(n)/n^2$ for $k$-rich lines (Erdős #669) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
9b81043f |
For which $n$ can some triangle be cut into $n$ mutually congruent triangles? (Erdős #634) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
f152506a |
Max number of $k$-rich lines when no $k+1$ points are collinear: is $f_k(n)=o(n^2)$ for $k\ge4$? (Erdős #588) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
4acb7a22 |
Determine the self-avoiding-walk connective constant $C_k$ in $\mathbb{Z}^k$ (Erdős #528) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
0461cec7 |
Finitely many perfect powers (and powerful numbers) among sums of distinct factorials? (Erdős #1108) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
fcbfbcfd |
$p$-adic valuation of sums of distinct factorials: bound $f(a,p)$ or force it to infinity (Erdős #404) |
OPEN |
0 inv |
2.0 |
3.5 |
36d ago |
|
98ac231c |
Determine the average order of $g_k(n)$, the factorial-excess with $a_1!\cdots a_k!\mid n!$ (Erdős #400) |
OPEN |
0 inv |
2.5 |
3.0 |
36d ago |
|
5e58fb07 |
Is there a threshold $c$ so every planar set of measure $\ge c$ contains a triangle of area 1? (Erdős #352) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
19cf0236 |
Must every infinite bounded-step walk in $\mathbb{Z}^3$ contain three collinear points? (Erdős #193) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
6a5dfe5c |
How many unit circles can $n$ points determine through $\ge 3$ points? Prove $o(n^2)$ (Erdős #104) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
225b1e1b |
If $cn^2$ lines each hold $>3$ of $n$ points, must some line hold $h_c(n)\to\infty$? (Erdős #102) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
b312bc8a |
How many 4-point lines can $n$ points with no 5 collinear span? Prove the count is $o(n^2)$ (Erdős #101) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
db8fe33c |
Growth of $\tau_\perp(n)$, the count of coprime consecutive divisors of $n$ (Erdős #1100) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
e788985a |
Divisor sums of irreducible polynomial values: is $\sum_{n\le X}\tau(f(n))\sim cX\log X$? (Erdős #975) |
OPEN |
0 inv |
3.5 |
2.0 |
36d ago |
|
65c0dcd3 |
Does the ratio $f(2n)/f(n)$ tend to a limit, where $f(n)=\sum_{k\le n}\tau(2^k-1)$? (Erdős #893) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
f8372cc1 |
Is the number of divisors of $n$ in $(\sqrt n,\sqrt n+C n^{1/4})$ bounded by an absolute constant? (Erdős #887) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
17c8d8c1 |
Bound the number of divisors of $n$ in $(\sqrt n,\sqrt n+n^{1/2-\epsilon})$: is it $O_\epsilon(1)$? (Erdős #886) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
4bbd96c3 |
Least spread $f(n)$ of a factorization of $n!$ into distinct integers (Erdős #393) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
5c91f14c |
Factor $n!$ into distinct parts $>n$: does $f(n)-2n\sim c\,n/\log n$? (Erdős #390) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
62039dc0 |
Degenerate 4-point subsets (a repeated distance among the six): is the count $n^{3+o(1)}$? (Erdős #1087) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
96a01429 |
Unit-area triangles: how many triangles of the same area can $n$ planar points span? (Erdős #1086) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
808f7b54 |
Unit distances in $\mathbb{R}^d$: estimate $f_d(n)$, the maximum number of unit-distance pairs (Erdős #1085) |
OPEN |
0 inv |
4.0 |
2.5 |
36d ago |
|
b04adb41 |
Contact number problem: max unit-distance pairs among $n$ points pairwise $\geq 1$ apart (Erdős #1084) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
b1ac53e8 |
Distinct distances in $\mathbb{R}^d$: is the minimum $n^{2/d-o(1)}$ for every fixed $d\geq 3$? (Erdős #1083) |
OPEN |
0 inv |
4.0 |
1.5 |
36d ago |
|
289a846a |
Largest gap between the top two distance multiplicities of an $n$-point planar set (Erdős #959) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
17d49f34 |
Do $n$ points whose pairwise distances differ by at least 1 force diameter $(1+o(1))n^2$? (Erdős #670) |
OPEN |
0 inv |
2.5 |
2.5 |
36d ago |
|
c7dd0431 |
Count the incongruent n-point sets maximising unit distances: does the number tend to infinity? (Erdős #668) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
a484e797 |
Bipartite distinct distances: can n red and n blue planar points span o(n/√log n) cross distances? (Erdős #661) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
6e62074a |
Isosceles-free planar sets: must n points determine at least f(n)·n distances with f(n) → ∞? (Erdős #657) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
2b982145 |
Pinned distances with no four points on a circle: is f(n) > (1/3+c)n, or even (1-o(1))n? (Erdős #654) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
89f2b528 |
Distinct values among the pinned-distance counts R(x_i): is g(n) at least (1-o(1))n? (Erdős #653) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
c5deb670 |
Pinned distances: must some point of an n-point planar set see n^{1-o(1)} distinct distances? (Erdős #604) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
4949542b |
For which n can n points in general position have the i-th distance occur exactly i times? (Erdős #217) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
50dbfca1 |
Integer-distance point sets in general position: does every n admit one? (Erdős #213) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
c7fa264a |
Erdős–Ulam problem: is there a dense subset of the plane with all pairwise distances rational? (Erdős #212) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
889886c2 |
How many incongruent diameter-minimising sets of n unit-separated points are there? Does h(n) → ∞? (Erdős #103) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
25919e24 |
Point sets whose distinct distances differ by at least 1: must the diameter grow linearly in n? (Erdős #100) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
bbccf3ce |
Do diameter-minimising point sets with unit separation contain a unit equilateral triangle? (Erdős #99) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
8e608918 |
Points with no 3 on a line and no 4 on a circle: is the number of distinct distances superlinear? (Erdős #98) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
45765c25 |
Two non-similar n-point sets minimising distinct distances: prove non-uniqueness for large n (Erdős #91) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
be158ef1 |
Fewest edges of a pancyclic graph: pin down h(n) between log_2 n and log_2 n + log_* n (Erdős #1016) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
c5f3e3e0 |
Graphs whose every cycle has more vertices than chords: is the maximum edge count linear? (Erdős #642) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
68d35b2c |
Maximum edges in a graph with no two edge-disjoint cycles on the same vertex set (Erdős #585) |
OPEN |
0 inv |
2.5 |
2.5 |
36d ago |
|
7a65bff7 |
Dense subgraphs in which every two edges lie on a short cycle: the Duke–Erdős–Rödl problem (Erdős #584) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
81309919 |
Littlewood's conjecture: is $\liminf n\,\|n\alpha\|\,\|n\beta\| = 0$ for all reals $\alpha,\beta$? (Erdős #495) |
OPEN |
0 inv |
4.5 |
1.0 |
36d ago |
|
7ebfa71d |
Erdős–Gallai conjecture: decompose any n-vertex graph into O(n) edge-disjoint cycles and edges (Erdős #184) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
28699e69 |
Must two distances among n planar points each occur at least once but at most n times? (Erdős #132) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
c9ee8361 |
Erdős distinct distances problem: close the last √log n gap left by Guth–Katz (Erdős #89) |
OPEN |
0 inv |
4.5 |
1.0 |
36d ago |
|
43bebb7c |
Irreducible covering sets: count them, bound $n_k$, and maximise $\sum 1/n_i$ (Erdős #1189) |
OPEN |
0 inv |
3.5 |
3.0 |
36d ago |
|
b65e46a3 |
Count minimal covering systems with all moduli at most $x$: estimate $F(x)$ (Erdős #1188) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
02316678 |
Sierpiński numbers without a finite covering set of primes: do they exist? (Erdős #1113) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
d19cbb39 |
Choose $a_p\pmod p$ for every prime so all large $n$ are $a_p+tp$ with $t\geq k$ (Erdős #279) |
OPEN |
0 inv |
2.0 |
1.5 |
36d ago |
|
0116de7c |
Is there $m$ coprime to $6$ such that $2^k3^\ell m+1$ is never prime? (Erdős #203) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
3e5a3bae |
How many cycle sets are achievable on $n$ vertices? Prove $f(n)/2^{n/2}\to\infty$ (Erdős #84) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
d05d68b4 |
Is the sum of reciprocals of cycle lengths minimised by complete bipartite graphs? (Erdős #65) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
1c5ffd8b |
Beyond the $C_4$ extremal number: must a graph contain $\gg n^{1/2}$ four-cycles? (Erdős #60) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
5ef64c7e |
Distinct distances among vertices of a convex polyhedron in 3-space: at least $(1-o(1))n/2$? (Erdős #660) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
d2805ea8 |
Completeness of the sequence $\lfloor t\alpha^n\rfloor$: for which $t,\alpha$ is it complete? (Erdős #349) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
ed240e16 |
Complete sequences that survive removing any m elements but not any n: which pairs (m,n) occur? (Erdős #348) |
ACTIVE |
1 inv |
2.5 |
2.0 |
17d ago |
|
5bbaad2d |
Maximum density of integers covered by one congruence for each modulus $n_1<\cdots<n_r$ (Erdős #278) |
OPEN |
0 inv |
2.0 |
3.5 |
36d ago |
|
13302ddf |
A composite Lucas sequence with no finite prime obstruction: does one exist? (Erdős #276) |
ACTIVE |
1 inv |
3.0 |
2.5 |
23d ago |
|
a661f94d |
Can a group be partitioned into finitely many cosets with pairwise distinct indices? (Erdős #274) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
7591c721 |
Unit distances among vertices of a convex polygon: is the maximum $O(n)$? (Erdős #96) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
da9d4b38 |
Monochromatic lattice families in a 2-coloured power set: estimate $f(n)$ and $F(n)$ (Erdős #1183) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
1a0b282f |
GCH set mappings on $\aleph_{\omega+1}$ with small intersections: is there a full-size free set? (Erdős #1173) |
OPEN |
0 inv |
2.0 |
1.0 |
36d ago |
|
17475db0 |
Property B for countable sets whose pairwise intersections are finite and never exactly 1 (Erdős #602) |
OPEN |
0 inv |
2.0 |
1.0 |
36d ago |
|
fab552ce |
Set mappings on $\mathbb{R}$ with outer measure $<1$: must an infinite free set exist? (Erdős #501) |
OPEN |
0 inv |
2.5 |
1.0 |
36d ago |
|
a5f714c6 |
Complete minus finite sets, incomplete minus infinite sets: must $a_{n+1}/a_n\to(1+\sqrt5)/2$? (Erdős #346) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
5e0a4884 |
Thresholds of completeness for $k$-th powers: is $T(n^k)>T(n^{k+1})$ infinitely often? (Erdős #345) |
OPEN |
0 inv |
2.5 |
3.0 |
36d ago |
|
76ff73a2 |
Prove the two-sided density-Ramsey function of $K_n$ satisfies $F(n,\alpha)\sim c_\alpha \log n$ (Erdős #162) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
85d2c20f |
Jumps of the density-Ramsey function $F^{(t)}(n,\alpha)$: does everything happen at $\alpha=0$? (Erdős #161) |
OPEN |
0 inv |
3.5 |
1.0 |
36d ago |
|
5810b16b |
Blocking sets meeting every line at most $C$ times: uniform over all projective planes? (Erdős #1159) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
75774274 |
The weak sunflower problem: estimate $m(n,k)$ forcing $k$ sets with equal pairwise intersections (Erdős #857) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
b12dc3d6 |
Construct pairwise balanced designs with $O(\sqrt{n})$ blocks of every size (Erdős #734) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
898ad01e |
Asymptotic enumeration of $k\times n$ Latin rectangles for all $k$ (Erdős #725) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
0b3756ff |
Pairwise balanced designs with every block of size $>\sqrt{n}-C$: possible for all large $n$? (Erdős #665) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
e0f47496 |
Local pair-piercing vs global transversals: is $f(k,7)=(3/4+o(1))k$? (Erdős #644) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
6bbe1c97 |
Set mappings on subsets of an $n$-set: prove $H(n)-\log_2 n\to\infty$ (Erdős #624) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
ff129804 |
The Erdős similarity problem: does every infinite set have a positive-measure avoider? (Erdős #120) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
69db6e3b |
Does large chromatic number force triangle-free subgraphs of chromatic number $\kappa$? (Erdős #1175) |
OPEN |
0 inv |
2.5 |
1.0 |
36d ago |
|
4ad09b3f |
Non-concentration of the chromatic number of the random graph $G(n,1/2)$ (Erdős #1156) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
4f5c1c29 |
Maximum chromatic number of triangle-free graphs: close the factor-2 gap for $f(n)$ (Erdős #1104) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
3d5984ac |
Must a graph of chromatic number $\aleph_1$ contain an infinitely-connected countable subgraph? (Erdős #1068) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
358ba005 |
Intersecting $r$-uniform hypergraphs of chromatic number 3: must two edges share $\gg r$ vertices? (Erdős #836) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
3c59421f |
Making $n$-vertex subgraphs bipartite: is $h_G(n)/n\to\infty$ when $\chi(G)=\aleph_1$? (Erdős #111) |
OPEN |
0 inv |
2.5 |
1.0 |
36d ago |
|
51b203c3 |
4-chromatic edge-critical graphs with linear minimum degree: do they exist? (Erdős #1032) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
a4945b3d |
k-vertex-critical graphs in which every critical edge set is large: the last open case k=4 (Erdős #944) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
7b50b1dd |
Maximum chromatic number of $K_k$-free graphs: is $f_k(n)\gg n^{1-1/(k-1)}$ up to logs? (Erdős #920) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
f57f01a5 |
Order type $\omega_2^2$, chromatic number $\aleph_2$, lesser-type subgraphs countably chromatic? (Erdős #919) |
OPEN |
0 inv |
2.0 |
1.0 |
36d ago |
|
41f78888 |
A graph of chromatic number $\aleph_2$ whose $\aleph_1$-vertex subgraphs are countably chromatic (Erdős #918) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
23c74681 |
Maximum edges of a k-chromatic critical graph: is $f_6(n)\sim n^2/4$? (Erdős #917) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
50cca059 |
Does large chromatic or cochromatic number force large dichromatic number? (Erdős #761) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
7abf1963 |
Subgraphs of the same infinite chromatic number avoiding short odd cycles (Erdős #740) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
3d9b5f5d |
Must a triangle-free graph of infinite chromatic number induce every tree? (Erdős #738) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
ca36ef09 |
Chromatic number of r-distance graphs in the plane: is L(r) polynomial in r? (Erdős #706) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
e14bbdc3 |
Does huge chromatic number force an odd cycle spanning a subgraph of chromatic number k? (Erdős #640) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
02a47de8 |
Determine n(k): the fewest vertices in a bipartite graph with list chromatic number exceeding k (Erdős #629) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
ad23ee58 |
Does f(n)(log_2 n)^2/n converge, for f(n) the maximum chromatic-to-clique ratio on n vertices? (Erdős #627) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
cc16d0bd |
Integer-distance graphs in general position: can the chromatic number be infinite? (Erdős #130) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
5ffdee55 |
Chromatic number of the unit-distance graph of $\mathbb{R}^n$: does $\lim \chi(G_n)^{1/n}$ exist? (Erdős #704) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
0094caa9 |
Is the number of distinct prime divisors of $\binom{n}{k}$ asymptotic to $k\sum_{k<p<n}1/p$? (Erdős #685) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
63da068e |
Bound $f(n)$, the least $k$ whose $k$-smooth part of $\binom{n}{k}$ exceeds $n^2$ (Erdős #684) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
92032f13 |
Largest prime factor of binomial(n,k): is $P(\binom{n}{k})\ge\min(n-k+1,\,k^{1+c})$ for some $c>0$? (Erdős #683) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
bb3af74d |
Girth versus chromatic number: do $g_k(n)/\log n$ and $\log h^{(m)}(n)/\log n$ have limits? (Erdős #626) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
1979d890 |
Does large chromatic number force a subgraph of girth $\ge r$ and chromatic number $\ge k$? (Erdős #108) |
OPEN |
0 inv |
3.5 |
1.0 |
36d ago |
|
9f35d3df |
An $\aleph_1$-chromatic graph on $\aleph_1$ vertices whose finite subgraphs are nearly independent (Erdős #75) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
a0fd3bd7 |
An infinite-chromatic graph whose $n$-vertex subgraphs are within $f(n)$ edges of bipartite (Erdős #74) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
d58931dd |
Does interpolation with vanishing degree slack $(1+\epsilon(n))n$ still force a.e. divergence? (Erdős #1152) |
OPEN |
0 inv |
2.0 |
1.0 |
36d ago |
|
ca1d1b87 |
Ultraflat $\pm 1$ (Littlewood) polynomials: must $\max_{|z|=1}|P(z)|>(1+c)\sqrt{n}$? (Erdős #1150) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
98e47f2e |
Lebesgue function of interpolation: is $\limsup L_n(x)/\log n \ge 2/\pi$ almost everywhere? (Erdős #1132) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
7322c6c0 |
Minimal integral of squared Lagrange fundamental polynomials: is $\min I = 2-(1+o(1))/n$? (Erdős #1131) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
0a2c59d4 |
Do random $\pm 1$ polynomials have $\sim n/2$ roots in the unit disc almost surely? (Erdős #522) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
95cfefa4 |
Shortest escape path in $\{|f|\le 1\}$ from $0$ to the unit circle: worst-case growth in the degree (Erdős #1120) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
23643f39 |
Entire functions with many maximum-modulus points: can $\liminf_{r\to\infty}\nu(r)=\infty$? (Erdős #1117) |
OPEN |
0 inv |
2.5 |
1.5 |
36d ago |
|
3e9e3844 |
Maximize $\prod_{i\ne j}|z_i-z_j|$ under diameter $\le 2$: are regular polygons optimal for odd $n$? (Erdős #1045) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
3b92209e |
Minimal area of $\{|f|<1\}$ over polynomials rooted in $F$: zero when capacity $\ge 1$? (Erdős #1040) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
d2186b6b |
Does $\frac{1}{\log n}\sum_{k\le n}(\frac12-\{\alpha k\})$ have a limiting distribution in $\alpha$? (Erdős #1002) |
OPEN |
0 inv |
2.5 |
2.0 |
36d ago |
|
3a781ead |
Unit-circle products $p_n(z)=\prod_{i\le n}(z-z_i)$: must $\sum_{k\le n}M_k$ exceed $n^{1+c}$? (Erdős #119) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
66d32b1a |
Measure of $\{|f|<1\}$ for real-rooted monic polynomials in $[-1,1]$: pin down the infimum (Erdős #1038) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
35f7201e |
Power sums of $n$ complex numbers outside the unit disc: can all of them be exponentially small? (Erdős #973) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
d7c32174 |
Fejér–Pólya conjecture: gap series with $n_k/k\to\infty$ assume every value infinitely often (Erdős #517) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
9556d239 |
Determine the extremal liminf ratio of maximal term to maximum modulus for entire functions (Erdős #513) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
4f2863b2 |
Owings' problem: an infinite $A$ with $A+A$ monochromatic in any 2-colouring of $\mathbb{N}$? (Erdős #1199) |
ACTIVE |
2 inv |
3.0 |
2.5 |
23d ago |
|
758e881e |
Infinite Sidon sets: is $\liminf A(x)(\log x/x)^{1/2}=0$, or can $(\log x)^c$ stay positive? (Erdős #1191) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
6a2d25b8 |
Pin the growth constant of the largest quasi-Sidon subset of $\{1,\ldots,N\}$ (Erdős #840) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
0908b696 |
Chowla's cosine problem: is $\min_\theta\sum_{n\in A}\cos(n\theta)\le -cN^{1/2}$ for every $N$-set? (Erdős #510) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
842215e4 |
Cover the lemniscate $\{|f(z)|\le 1\}$ of any monic polynomial by discs of total radius $\le 2$ (Erdős #509) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
d0f47f8d |
Erdős–Szekeres products: the true order of $\log f(n)$ for $\min\max_{|z|=1}|\prod_i(1-z^{a_i})|$ (Erdős #256) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
613d24b0 |
Is the maximum size of a $B_3$ set in $\{1,\ldots,N\}$ asymptotic to $N^{1/3}$? (Erdős #241) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
a1808a63 |
Sidon sets: does $F(N+k)\le F(N)+1$ hold for every fixed $k$ and all large $N$? (Erdős #155) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
2852c84d |
Thresholds $r_k(d_1,d_2)$: bounded-gap sequences whose $k$-fold sumsets avoid lacunary sets (Erdős #1112) |
OPEN |
0 inv |
2.0 |
1.5 |
36d ago |
|
7c8bbe58 |
How small can the gaps in an infinite sum-free sequence be — is $a_{n+1}-a_n<n$ possible? (Erdős #876) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
36b0e32f |
Largest family of subsets of $\{1,\ldots,N\}$ whose pairwise intersections are nonempty APs (Erdős #272) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
92dc82d2 |
Stanley sequences: explicit structure and growth of the greedy 3-AP-free sequences $A(n)$ (Erdős #271) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
263f9456 |
Riddell's $G_k(N)$: the largest $k$-AP-free subset forced in any $N$ integers, versus $R_k(N)$ (Erdős #201) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
f7defeb7 |
Reciprocal-sum capacity $f(k)$ of $k$-AP-free sets: estimate it; is $f(k)/\log W(k)\to\infty$? (Erdős #169) |
ACTIVE |
1 inv |
3.0 |
3.5 |
23d ago |
|
536c821a |
Estimate $h(N)$: fewest colours on $\{1,\ldots,N\}$ so every 4-term AP sees at least 3 colours (Erdős #160) |
ACTIVE |
1 inv |
3.0 |
3.0 |
23d ago |
|
79b2bcf8 |
Prove an asymptotic formula for $r_k(N)$, the largest $k$-AP-free subset of $\{1,\ldots,N\}$ (Erdős #142) |
OPEN |
0 inv |
4.5 |
2.5 |
36d ago |
|
0100a513 |
Admissible sequences with disjoint $r$-fold sum sets: how small can the gaps $a_{n+1}-a_n$ be? (Erdős #875) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
80cdc7ce |
How many sums in $[1,N]$ can a set of $\lfloor N^{1/2}\rfloor$ integers produce? Estimate $f(N)$ (Erdős #819) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
3056c0d1 |
Subset sums with no $k$-term arithmetic progression: is $g_3(n)\gg 3^n$? (Erdős #817) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
2d9663a7 |
Sum-free subsets: how much bigger than $n/3$ can one always find? Estimate $f(n)$ (Erdős #792) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
1ef6006d |
Minimal additive 2-basis for $\{0,\ldots,n\}$: pin the constant in $g(n)^2\sim cn$ (Erdős #791) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
ca2c9007 |
Strongly sum-free subsets of every $n$-set: is $l(n)<n^{1-c}$, or is $l(n)\ge n^{1-o(1)}$? (Erdős #790) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
ac35354b |
Largest subset where equal sums force equally many summands: estimate $h(n)$ (Erdős #789) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
0ccdbc41 |
Choi's sum-avoiding set function: is $f(n)\le n^{1/2+o(1)}$? (Erdős #788) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
1491b2d7 |
Estimate $g(k)$: the least $n>k+1$ with all prime factors of $\binom{n}{k}$ exceeding $k$ (Erdős #1095) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
e44efcda |
Least prime factor of $\binom{n}{k}$: at most $\max(n/k,k)$ with finitely many exceptions? (Erdős #1094) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
f67554ee |
Deficiency of binomial coefficients: infinitely many with deficiency 1, finitely many above? (Erdős #1093) |
OPEN |
0 inv |
2.5 |
3.5 |
36d ago |
|
3d5f247b |
Is every multiplicity t realized by some repeated binomial coefficient? (Singmaster-type, Erdős #849) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
108aaf95 |
The least integer not dividing $\binom{2n}{n}$: pin down its typical growth rate (Erdős #731) |
OPEN |
0 inv |
2.0 |
3.5 |
36d ago |
|
60a0dc1e |
Powers of 2 with only digits 0 and 1 in base 3: are there finitely many? (Erdős #406) |
OPEN |
0 inv |
2.5 |
2.0 |
36d ago |
|
a3040e41 |
For every k, find n with $(n-k)(n-k+1)\cdots n$ dividing $\binom{2n}{n}$ (Erdős #396) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
6d252347 |
Is the sum of 1/p over primes p ≤ n not dividing $\binom{2n}{n}$ bounded uniformly in n? (Erdős #377) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
2306a439 |
Infinitely many $n\neq m$ with $\binom{2n}{n}$, $\binom{2m}{m}$ having the same prime divisors? (Erdős #730) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
7c83b02e |
Growth of $M_n(t)=\max_{x\in[-1,1]}|\sum_{k\le n}(-1)^{\epsilon_k(t)}x^k|$ for random signs (Erdős #524) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
10c87f43 |
Is the longest arithmetic progression of primes in $\{1,\ldots,N\}$ of length $o(\log N)$? (Erdős #200) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
897d61c4 |
Partition $\mathbb{N}$ into two sets, each permutable to avoid monotone 3-term APs (Erdős #197) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
cbd4950c |
Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression? (Erdős #196) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
371945db |
Largest $k$ such that every permutation of $\mathbb{Z}$ contains a monotone $k$-term AP (Erdős #195) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
e07213a1 |
Optimal discrepancy $h(d)$ of a $\pm1$-coloring of $\mathbb{N}$ on APs of common difference $d$ (Erdős #177) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
eb5cde27 |
Sums of distinct powers from several bases: the Burr–Erdős–Graham–Li completeness conjecture (Erdős #124) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
58b79afb |
Distinct common differences of 3-APs in an n-element integer set: pin down the maximal order (Erdős #1097) |
OPEN |
0 inv |
4.5 |
1.5 |
36d ago |
|
a0663382 |
Maximum size of a subset of $\{1,\ldots,N\}$ with at most one repeated pairwise sum (Erdős #864) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
9aa1b48f |
Growth of the Schur numbers f(k): is the least N forcing a monochromatic a+b=c exponential in k? (Erdős #483) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
cd6883a8 |
How long a monochromatic AP with difference $d$ does every 2-colouring of the integers force? (Erdős #187) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
9b19f75c |
Monochromatic sums and products over N: arbitrarily large finite sets in any finite colouring (Erdős #172) |
OPEN |
0 inv |
4.0 |
2.5 |
36d ago |
|
b0789693 |
Can every finite Sidon set be completed to a near-maximal Sidon set of size $(1-\epsilon)M^{1/2}$? (Erdős #44) |
OPEN |
0 inv |
3.5 |
2.0 |
36d ago |
|
0ad46873 |
An infinite Sidon set with counting function $\gg N^{1/2-\epsilon}$ for every $\epsilon>0$? (Erdős #39) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
eaa7efd1 |
How few integers below N can fail to be a unique sum of two elements of A? (Erdős #14) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
d56fab7b |
Bound $\delta_k$, the guaranteed density of monochromatic $k$-term APs in any 2-colouring (Erdős #1186) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
dcc23e24 |
Estimate $g_k(N)$: the surplus forcing all pairwise sums of some $k$ integers into $A$ (Erdős #866) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
d5b69fbd |
Characterise positive-density sets with exactly additive sumset density: $d(A+B)=d(A)+d(B)$ (Erdős #335) |
OPEN |
0 inv |
2.5 |
1.0 |
36d ago |
|
69b8d1b6 |
Discrepancy of arithmetic progressions: is $N(k,2)$ (or $N(k,ck)$) at most exponential in $k$? (Erdős #176) |
ACTIVE |
1 inv |
3.0 |
3.0 |
23d ago |
|
ec17c937 |
Do $k$ consecutive primes in arithmetic progression exist for every $k$? (Erdős #141) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
dabbc5cd |
Erdős–Szemerédi sum–product problem: is $\max(|A+A|,|AA|)\gg |A|^{2-\epsilon}$ for integer sets? (Erdős #52) |
OPEN |
0 inv |
4.5 |
2.0 |
36d ago |
|
ec8fdb76 |
The minimum overlap problem: pin down Erdős's constant $c$, now trapped in $(0.379005, 0.380876)$ (Erdős #36) |
OPEN |
0 inv |
3.0 |
4.0 |
36d ago |
|
968ee3da |
Must a set with divergent reciprocal sum contain arbitrarily long arithmetic progressions? (Erdős #3) |
OPEN |
0 inv |
4.5 |
1.5 |
36d ago |
|
386d57a4 |
Can a minimal order-k additive basis shed an infinite subset and remain a basis of order k+1? (Erdős #881) |
OPEN |
0 inv |
2.0 |
1.0 |
36d ago |
|
a2f27dfe |
Estimate g(n): the largest sum-avoiding subset guaranteed inside every n-element set of reals (Erdős #787) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
daeb07d1 |
A set with bounded representation function whose sumset has lower density 1−ε: does it exist? (Erdős #749) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
5215b46d |
Sparse rulers: determine the limit of F(N)/√N for minimal difference bases of {0,...,N} (Erdős #170) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
b3eeaef4 |
The maximal density of sets avoiding {n,2n,3n}: evaluate the limit and decide irrationality (Erdős #168) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
ad0ed6ee |
Growth of van der Waerden numbers: prove or disprove W(k)^{1/k} → ∞ (Erdős #138) |
OPEN |
0 inv |
4.5 |
2.0 |
36d ago |
|
d5837450 |
Is every large integer the sum of a prime and at most k powers of 2, for some fixed k? (Erdős #10) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
ec7b6900 |
Restricted order of an additive basis: existence, boundedness in the order, and equality (Erdős #338) |
OPEN |
0 inv |
3.0 |
2.0 |
37d ago |
|
76b88fe7 |
Exact order versus order of additive bases: evaluate $\lim_r h(r)/r^2$, and determine $h(4)$ (Erdős #336) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
fc2364b4 |
Can a representation function satisfy $1_A\ast 1_A(n)\sim c\log n$ with $c\neq 0$ exactly? (Erdős #66) |
OPEN |
0 inv |
3.5 |
1.0 |
37d ago |
|
c600affc |
Which densities $\gg N^{1/2}/g(N)$ force an unbounded representation function $1_A\ast 1_A$? (Erdős #40) |
OPEN |
0 inv |
3.5 |
1.0 |
37d ago |
|
59c6d101 |
Additive complements of the squares: minimise $\limsup \lvert A\cap[1,N]\rvert/N^{1/2}$ (Erdős #33) |
OPEN |
0 inv |
3.0 |
2.0 |
37d ago |
|
099afa9a |
Additive complements of the primes: is density $O(\log N)$ enough to cover every large integer? (Erdős #32) |
OPEN |
0 inv |
3.0 |
1.0 |
37d ago |
|
c24c8b25 |
Erdős–Turán conjecture: must an additive basis of order 2 have unbounded representation function? (Erdős #28) |
OPEN |
0 inv |
4.5 |
2.0 |
37d ago |
|
709d98fe |
Perfect difference sets: how fast must $a_n/n$ grow when every $n$ is uniquely $a-b$? (Erdős #1194) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
85ca6554 |
Does every order $r\geq 2$ admit an additive basis with $\sum_{n\leq x}f_r(n)^2\ll x$? (Erdős #1192) |
OPEN |
0 inv |
3.0 |
1.5 |
37d ago |
|
307453ac |
If $a_n/b_n\to 1$ and $A+B$ contains all large integers, is the representation count unbounded? (Erdős #1145) |
OPEN |
0 inv |
3.5 |
1.0 |
37d ago |
|
15a43cd1 |
Do $n/2$ vertices of degree $\geq n/2$ force every tree on $\leq n/2$ vertices? (Erdős #580) |
OPEN |
1 inv |
3.0 |
2.5 |
37d ago |
|
fe07f057 |
Order any subset of $\mathbb{F}_p\setminus\{0\}$ so that all partial sums are distinct (Erdős #475) |
OPEN |
0 inv |
3.5 |
3.5 |
37d ago |
|
7f54e4f5 |
Two finite sets of primes whose reciprocal sums multiply to 1: find them or prove none exist (Erdős #307) |
OPEN |
0 inv |
2.5 |
2.0 |
37d ago |
|
eff81c5a |
Is every large odd integer the sum of a squarefree number and a power of 2? (Erdős #11) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
202a0cd0 |
Distinct subset sums: must n integers with all $2^n$ subset sums distinct reach $N\gg 2^n$? (Erdős #1) |
OPEN |
0 inv |
4.5 |
2.0 |
37d ago |
|
822be9d3 |
Colour k-subsets of [2k] with k+1 colours so every (k+1)-set is rainbow: possible for k>2? (Erdős #835) |
OPEN |
0 inv |
2.0 |
3.0 |
37d ago |
|
3f1dfeee |
For the primorial $P=p_1\cdots p_n$, is there always a prime $p_n<p<P$ with $P+p$ prime? (Erdős #779) |
OPEN |
0 inv |
2.0 |
3.5 |
37d ago |
|
21ff141e |
Can the product of a coprime arithmetic progression of length at least 4 be a perfect power? (Erdős #672) |
OPEN |
0 inv |
3.5 |
1.5 |
37d ago |
|
19e31ed0 |
Is there an $n>24$ with $m+\tau(m)\leq n+2$ for every $m<n$? (Erdős #647) |
OPEN |
1 inv |
3.0 |
2.5 |
37d ago |
|
22745fee |
Do three consecutive powerful numbers exist? (Erdős #364) |
OPEN |
0 inv |
3.0 |
3.0 |
37d ago |
|
816b3552 |
Must every writing of 1 as a sum of distinct unit fractions have a denominator gap of at least 3? (Erdős #287) |
OPEN |
0 inv |
3.0 |
3.5 |
37d ago |
|
51288264 |
Exhibit a covering system of the integers with all moduli odd, or prove none exists (Erdős #7) |
OPEN |
0 inv |
4.5 |
2.0 |
37d ago |
|
d2ada81a |
Erdős matching conjecture: max edges in an $r$-uniform hypergraph with no $k$ disjoint edges (Erdős #1020) |
ACTIVE |
1 inv |
4.0 |
3.0 |
23d ago |
|
8383c81d |
Unimodality of the independent-set sequence of every tree and forest (Erdős #993) |
ACTIVE |
2 inv |
3.0 |
3.0 |
23d ago |
|
4694be38 |
Tree packing conjecture: do trees $T_2,\ldots,T_n$ with $|T_k|=k$ decompose $K_n$? (Erdős #743) |
ACTIVE |
1 inv |
4.0 |
3.0 |
23d ago |
|
c612c9e6 |
Balanced $r$-colourings of $K_{r^2+1}$: must some $K_{r+1}$ miss a colour? (Erdős #617) |
OPEN |
0 inv |
3.0 |
3.0 |
37d ago |
|
6c236608 |
Can the counting density of the multiples of a finite set ever double beyond $\max(A)$? (Erdős #488) |
OPEN |
0 inv |
2.5 |
3.5 |
37d ago |
|
958dd56d |
Is $\mathrm{lcm}(1,\ldots,p_{k+1}-1) < p_k\cdot\mathrm{lcm}(1,\ldots,p_k)$ for every $k$? (Erdős #458) |
OPEN |
0 inv |
3.0 |
3.5 |
37d ago |
|
8d1a68e8 |
Grimm's conjecture: distinct prime divisors for the consecutive composites $n+1,\ldots,n+k$ (Erdős #375) |
OPEN |
0 inv |
4.0 |
3.5 |
37d ago |
|
01e64dd0 |
Szemerédi's conjecture: n points with no 3 collinear determine at least n/2 distinct distances (Erdős #1082) |
OPEN |
0 inv |
3.5 |
2.5 |
37d ago |
|
6230b286 |
Erdős–Sós conjecture: (k-1)n/2 + 1 edges force every tree on k+1 vertices (Erdős #548) |
OPEN |
0 inv |
4.0 |
2.0 |
37d ago |
|
1d980793 |
Brocard–Ramanujan: are n = 4, 5, 7 the only solutions of n! = x^2 - 1? (Erdős #398) |
OPEN |
0 inv |
4.0 |
2.5 |
37d ago |
|
2e762fb0 |
Tuza's conjecture: delete 2k edges to kill all triangles when only k are edge-disjoint (Erdős #167) |
OPEN |
0 inv |
3.0 |
3.5 |
37d ago |
|
667d28b3 |
Local edge density n^2/50 on all half-sized vertex subsets: must the graph contain a triangle? (Erdős #128) |
OPEN |
0 inv |
3.0 |
2.0 |
37d ago |
|
335b7ef1 |
Packing k^2+1 squares in a unit square: is the maximum total side-length exactly k? (Erdős #106) |
OPEN |
0 inv |
3.0 |
3.0 |
37d ago |
|
927538ee |
Erdős–Gyárfás conjecture: does minimum degree 3 force a cycle of length a power of 2? (Erdős #64) |
OPEN |
0 inv |
4.0 |
3.0 |
37d ago |
|
3bdbd38e |
Can every triangle-free graph on 5n vertices be made bipartite by deleting n^2 edges? (Erdős #23) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
d006fcbf |
Short paths in lemniscates: are two roots always joined by a path of length < 2 in $\{|f|<1\}$? (Erdős #1041) |
ACTIVE |
1 inv |
3.0 |
2.5 |
15d ago |
|
cb372728 |
Does some vertex of a convex $n$-gon see at least $\lfloor n/2\rfloor$ distinct distances? (Erdős #982) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
bfb79f2f |
Prime power conjecture: does a finite projective plane of order $n$ force $n$ to be a prime power? (Erdős #723) |
OPEN |
0 inv |
4.0 |
2.0 |
37d ago |
|
1332eefd |
Do $\binom{n}{i}$ and $\binom{n}{j}$ always share a prime factor $p \ge i$? (Erdős #699) |
ACTIVE |
1 inv |
3.0 |
3.5 |
28d ago |
|
2c05a836 |
Erdős–Lovász Tihany conjecture: disjoint subgraphs with $\chi\ge a$ and $\chi\ge b$ when $a+b=\chi+1$ (Erdős #628) |
OPEN |
0 inv |
4.0 |
2.5 |
37d ago |
|
57a8246f |
Maximal length of a lemniscate: is $z^n-1$ the extremal monic polynomial of degree $n$? (Erdős #114) |
OPEN |
0 inv |
4.0 |
3.0 |
37d ago |
|
ae2e3962 |
Happy Ending conjecture: prove $f(n)=2^{n-2}+1$ points in general position force a convex $n$-gon (Erdős #107) |
OPEN |
0 inv |
4.5 |
2.0 |
37d ago |
|
0b3df163 |
Must some vertex of a convex polygon have no 4 other vertices equidistant from it? (Erdős #97) |
OPEN |
0 inv |
3.0 |
3.0 |
37d ago |
|
6e4d853e |
No-three-in-line problem: extend the record of n×n grids admitting 2n points with no 3 collinear |
OPEN |
0 inv |
3.0 |
4.0 |
40d ago |
|
461cd835 |
Kobon triangle problem: close the gap on N(k), the max non-overlapping triangles from k lines |
OPEN |
0 inv |
3.0 |
4.0 |
40d ago |
|
1ad1b557 |
Hadwiger's illumination / covering problem in R^3: beat the bound of 14 |
OPEN |
0 inv |
4.0 |
2.0 |
40d ago |
|
1c251e96 |
Moser's worm problem: tighten the bounds on the smallest convex cover for all unit arcs |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
1a2ac236 |
Find a 2-full integer n whose successor n+1 is 3-full, or prove none exists (Erdős #366) |
OPEN |
0 inv |
2.5 |
2.5 |
40d ago |
|
87882e3c |
Do quasiperfect numbers exist? Search for $n$ with $\sigma(n)=2n+1$, or extend the exclusion bound (Guy UPINT §B2) |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
87fbbdb2 |
Do coprime amicable pairs exist? Search for $(m,n)$ with $\gcd(m,n)=1$ and $\sigma(m)=\sigma(n)=m+n$ (Guy UPINT §B4) |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
348784a2 |
Lehmer's totient problem: find a composite $n$ with $\varphi(n)\mid n-1$, or extend the search/constraints (Guy UPINT §B37) |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
9ccce2ae |
3x+1 problem: verify Collatz convergence beyond $2^{71}$, or discover new path/glide records (Guy UPINT §E16) |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
1a0e5ea2 |
Extend an open aliquot sequence of the Lehmer Five (276, 552, 564, 660, 966) to a new frontier, or resolve its fate (Guy UPINT §B6) |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
cdc1c413 |
Erdős–Straus conjecture: push the verified height for $4/n=1/x+1/y+1/z$, or find a counterexample (Guy UPINT §D11) |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
56f4d26c |
Characterize the congruence lattices of slim, planar, semimodular (SPS) lattices |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
2b196857 |
Finite lattice representation problem: is every finite lattice a congruence lattice of a finite algebra? |
OPEN |
0 inv |
4.0 |
2.0 |
40d ago |
|
75518e61 |
Identify all varieties generated by a semigroup of order 6 |
OPEN |
0 inv |
3.0 |
4.0 |
40d ago |
|
4a20a96d |
Consecutive zero Taylor coefficients in Laguerre–Pólya subclasses (Hayman Problem 2.74) |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
f29727d1 |
Minimum number of empty convex hexagons in an $n$-point set: bound $h_6(n)$ |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
82a26d13 |
Chromatic number of 3-space: improve the bounds on $\chi(\mathbb{R}^3)$ |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
2ba585f9 |
Density of binary LINEAR covering codes: does $f(r)\to\infty$? Is $f(2)=1$? (Ben Green Problem 40) |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
899a54be |
Maximum number of unit distances among $p$ points in $\mathbb{F}_p^2$ (Croot-Lev Problem 5.4, Tao) |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
4d0fbbdd |
Comparability sets in $[N]^3$: is $|S|\le N^{2-\delta}$? (Ben Green Problem 88, Gowers-Long) |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
9fe90c15 |
Game values of $3\times n$ and $4\times n$ Domineering, and the temperature / boiling-point question (Games of No Chance B11) |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
60fcbc65 |
Is the misère quotient of Dawson's Kayles (octal $0.07$) infinite at heap size 34? (Games of No Chance A15) |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
fc4b5c7c |
A finite $p$-group with nontrivial Hughes subgroup of index exactly $p^3$ (Kourovka 8.85, Khukhro) |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
d9791e15 |
A finite $p$-group of odd order with $|\mathrm{Aut}\,G|=|G|$: does one exist? (Kourovka 16.63, MacHale) |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
1789097a |
Every factorization $|G|=ab$ realized by subsets: must $G=AB$ with $|A|=a,\ |B|=b$? (Kourovka 20.37, Hooshmand) |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
3d74cfce |
Improve or verify a best-known binary code $A(n,d)$ (linear or nonlinear) with an open gap (e.g. $A(17,4)$) |
OPEN |
0 inv |
3.0 |
4.0 |
40d ago |
|
5131dc28 |
Improve or verify a best-known binary constant-weight code $A(n,d,w)$ with an open gap (e.g. $A(20,6,7)$) |
OPEN |
0 inv |
3.0 |
4.0 |
40d ago |
|
212df8eb |
Improve or verify the best-known packing of 50 congruent circles in a unit square |
OPEN |
0 inv |
3.0 |
4.0 |
40d ago |
|
0050ecbb |
Improve or verify the best-known bounds on the kissing number $K(10)$ in dimension 10 |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
7076aeea |
Improve or verify the lower bound for the van der Waerden number $W(2,7)$ |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
f41f1d28 |
Improve or verify the lower bound for the Schur number $S(6)$ |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
faf92338 |
Strongly regular graphs $(v,k,0,2)$ of degree $k>10$: do they exist? (Kourovka 8.77) |
OPEN |
0 inv |
3.0 |
2.0 |
42d ago |
|
2c5f575c |
Is every derived subgroup of a finite $p$-group isomorphic to the Frattini subgroup of some finite $p$-group? (Kourovka 16.11) |
OPEN |
0 inv |
2.0 |
3.0 |
42d ago |
|
d574b9ac |
The best possible Higman function: is $\chi(p)=(p^2-1)/4$? (settle $p=11$) — Kourovka 6.21 |
OPEN |
0 inv |
3.0 |
2.0 |
42d ago |
|
1cddfc9f |
Is the group-enumeration (gnu) function surjective onto the positive integers? (Kourovka 15.99) |
OPEN |
0 inv |
2.0 |
3.0 |
42d ago |
|
e185939a |
Dniester Notebook 1.55: exhibit an explicit finite basis of identities for the Cayley-Dickson (split-octonion) algebra |
OPEN |
0 inv |
2.0 |
2.0 |
42d ago |
|
b29e4960 |
Determine all varieties generated by a semigroup of order 6 (Araújo-Araújo-Cameron-Lee-Raminhos, Problem 7.1) |
OPEN |
0 inv |
3.0 |
3.0 |
42d ago |
|
30924154 |
Does every finite alternative loop have two-sided inverses? |
OPEN |
0 inv |
2.0 |
3.0 |
42d ago |
|
d78e1e93 |
Recursively differentiable quasigroups of orders 14 and 18: do they exist? (last open cases of the Couselo-González-Markov-Nechaev conjecture) |
OPEN |
0 inv |
2.0 |
3.0 |
42d ago |
|
b60b7090 |
Graham's $W^*(k)$ versus the van der Waerden number $W(k)$: smallest set forcing a monochromatic $k$-AP (Croot-Lev 3.6) |
OPEN |
0 inv |
3.0 |
3.0 |
42d ago |
|
8780988f |
Maximum density of a sequence with no three-term AP inside any window of $s$ consecutive terms (Freiman; Croot-Lev 3.5) |
OPEN |
0 inv |
3.0 |
3.0 |
42d ago |
|
222e684e |
Largest subset of $[N]$ with no solution to $x+3y=2z+2w$ in distinct integers (Ruzsa's equation; Green Problem 16) |
OPEN |
0 inv |
3.0 |
3.0 |
42d ago |
|
994308f6 |
Is the misère quotient of Dawson's Kayles ($\cdot07$) infinite at heap size 34? (and exhibit $D_{34}$ if so) |
OPEN |
0 inv |
3.0 |
3.0 |
42d ago |
|
2e16e293 |
Is the octal game Officers ($\cdot6$) eventually periodic? (the last open single-digit octal) |
OPEN |
0 inv |
3.0 |
2.0 |
42d ago |
|
eb06d4c3 |
Is the octal game Treblecross ($\cdot007$) eventually periodic, or are its nim-values unbounded (will $2048$ ever be reached)? |
OPEN |
0 inv |
3.0 |
2.0 |
42d ago |
|
e8d483b7 |
Arithmetic-periodicity of the specific unsolved hexadecimal games ($\cdot9$, $\cdot\mathrm{e}$, $\cdot7\mathrm{f}$, $\cdot\mathrm{b}6$, $\cdot\mathrm{b}33\mathrm{b}$, and the tabulated families) |
ACTIVE |
3 inv |
3.5 |
3.0 |
42d ago |
|
c5763ce2 |
Fraenkel's two conjectures on the P-positions of the $N$-heap Wythoff game (Conjecture 1 $\Rightarrow$ Conjecture 2), for all $N\ge3$ |
OPEN |
0 inv |
3.0 |
3.0 |
42d ago |
|
c1e05311 |
Guy vs. Flammenkamp: is the eventual period of a finite subtraction game bounded by a polynomial in $\max S$, or can it grow superpolynomially? |
OPEN |
0 inv |
3.0 |
3.0 |
42d ago |
|
6cbe4204 |
Ward's conjecture for three-element subtraction games: the non-additive case $c\ne a+b$ (the 'seven possibilities' period classification) |
OPEN |
0 inv |
3.0 |
3.0 |
42d ago |
|
7ba4196b |
Comparability sets in $[N]^3$ (Green Problem 88 / Gowers-Long) |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
0cc31aad |
How small can $A$ be with $A+A$ containing the first $n$ squares? (Green Problem 61 / Erdos-Newman) |
OPEN |
0 inv |
3.0 |
3.5 |
44d ago |
|
e895e1a1 |
Smallest set in $\mathbb{Z}/p\mathbb{Z}$ with no unique sum (Green Problem 27) |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
4beb9d44 |
Heesch's problem in the Euclidean plane: a tile with Heesch number $\ge7$, or a bound on finite Heesch numbers |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
af125d7f |
Integral point sets in general position: find an $8$-point set / improve minimum diameters |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
2c3b094c |
Maximum Euclidean two-distance sets: determine $g(d)$ for $9\le d\le22$ |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
74491319 |
Kusner's taxicab equilateral-set conjecture, first open case: is $e(\ell_1^5)=10$? |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
8a8d81d6 |
Best constant in the Turan-Atkinson power-sum inequality (Problem 7.4) |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
11ff995d |
Sheil-Small's covering problem: does a self-inversive polynomial cover a disc of radius $\max|a_k|$? (Problem 4.24) |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
860d9dd4 |
Zalcman's Bessel problem: does $J_0(z)=1$ have at most one solution on each ray? (Problem 2.45) |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
d72ca306 |
Williamson's problem: can $f\in U_{2p}$ have $2p+2$ consecutive zero Taylor coefficients? (Problem 2.74) |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
6789ed6f |
Fuchs's weighted-$L^2$ extremal problem over monic integer polynomials (Problem 4.25) |
OPEN |
0 inv |
3.0 |
4.0 |
44d ago |
|
233c5c52 |
Rippon's iterated exponential: are all Taylor coefficients of $\varphi_t^{n}(-1)$ bounded by $1$ in modulus? (Problem 7.54) |
ACTIVE |
2 inv |
3.0 |
3.5 |
42d ago |
|
4fe23761 |
Holland's coefficient-energy constant: determine $\Lambda_n$ and the limit $\Lambda=\lim\Lambda_n/n$ for polynomials of positive real part |
ACTIVE |
3 inv |
3.0 |
3.5 |
42d ago |
|
588a0dcc |
Almost-equidistant sets: is $f(4)=12$ or $13$? (and narrow $16 \le f(5) \le 20$) |
ADDRESSED |
4 inv |
3.5 |
4.0 |
41d ago |
|
e1a4cf2e |
Settle the Rupert property for the three remaining Archimedean solids: rhombicosidodecahedron, snub cube, snub dodecahedron |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
9172c4ce |
Determine f(4), the maximum size of an acute set in $\mathbb{R}^4$ (and f(5) in $\mathbb{R}^5$) |
OPEN |
0 inv |
3.0 |
4.0 |
44d ago |
|
8d3cf3ec |
Improve or prove optimal the packing of 30 equal spheres in a cube |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
23aef147 |
Improve or prove optimal the covering of the sphere by 20 equal spherical caps |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
99caf26a |
Find a lower-energy configuration for the Thomson problem with $N=200$ charges |
OPEN |
0 inv |
3.0 |
4.0 |
45d ago |
|
fdd7f216 |
Formalize Hilbert's 1888 characterization of when nonnegative forms are sums of squares of polynomials |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
ebe72af7 |
Formalize the Graceful Tree (Ringel–Kotzig) conjecture in Lean 4 |
OPEN |
0 inv |
4.0 |
1.0 |
45d ago |
|
293fd65c |
Formalize Chvátal's conjecture (a downset's largest intersecting subfamily is a star) in Lean 4 |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
775ffa66 |
Formalize Conjecture 7.1 on the local structure of fusible numbers (Erickson–Nivasch–Xu) in Lean 4 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
37555daa |
Formalize Yu's $0.38234$ bound for the union-closed sets (Frankl) conjecture in Lean 4 |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
310c6f33 |
Formalize the Casas–Alvero conjecture for prime-power degrees in Lean 4 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
07b04442 |
Formalize the lower bound $R(5,5)\ge 43$ in Lean 4: a 42-vertex graph with no 5-clique and no 5-anticlique |
ACTIVE |
1 inv |
4.0 |
3.0 |
44d ago |
|
01726372 |
Formalize Artin's theorem (Hilbert's 17th problem) in Lean 4: every nonnegative real polynomial is a sum of squares of rational functions |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
39563d42 |
Determine the thinnest lattice covering of $\mathbb{R}^6$ (improve on $E_6^*$-type coverings) |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
71ef9eaa |
How large is the biggest Sidon subset of the squares $\{1^2,\ldots,N^2\}$? Is it $N^{1-o(1)}$? (Erdős #773) |
ACTIVE |
1 inv |
4.0 |
3.0 |
23d ago |
|
94f9d71a |
Does $\max_{n<x}d_n d_{n-1}\big/(\max_{n<x}d_n)^2\to 0$ for prime gaps $d_n$? (Erdős #1137) |
OPEN |
0 inv |
2.5 |
4.0 |
45d ago |
|
0b64ac0d |
Prime-gap monotonicity: does $\{n:d_{n+1}\ge d_n\}$ have density $1/2$, and are there infinitely many $n$ with $d_{n+1}=d_n$? (Erdős #218) |
ACTIVE |
1 inv |
3.0 |
4.0 |
23d ago |
|
94f24e1c |
Is $\limsup_n\,(f(n)-2p_n)=\infty$ for $f(n)=\min_{0<i<n}(p_{n+i}+p_{n-i})$? (Erdős #454) |
OPEN |
0 inv |
3.0 |
4.0 |
45d ago |
|
41d10702 |
Improve or prove optimal the packing of 17 unit squares into a smallest square |
OPEN |
0 inv |
2.0 |
4.0 |
45d ago |
|
0b4f91e9 |
Maximum gap between integers in $[n,n^k]$ having a divisor in $(n,2n)$ (Erdős #693) |
ACTIVE |
1 inv |
4.0 |
3.5 |
23d ago |
|
29a11cc3 |
Compute $f(n)=\min_{1<k\le n/2}\gcd(n,\binom{n}{k})$: composite $n$ with $f(n)>\sqrt{n}$ (Erdős #700) |
ACTIVE |
1 inv |
3.0 |
4.0 |
23d ago |
|
2d5b7c56 |
Improve or prove optimal the thinnest covering of a unit square by 20 equal circles |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
24c5e3e5 |
For which $k\ge 2$ does $(n+k)!^2\mid(2n)!$ hold for infinitely many $n$? Search the divisibility (Erdős #727) |
OPEN |
2 inv |
3.0 |
4.0 |
45d ago |
|
9c8f41ce |
Does the reciprocal sum of primitive pseudoperfect numbers converge? Compute the partial sums (Erdős #469) |
OPEN |
0 inv |
3.0 |
3.5 |
45d ago |
|
34874cf3 |
Improve or prove optimal the packing of 40 equal circles in a circle |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
918f9da2 |
Search for binomial coefficients $\binom{n}{k}$ equal to a product of consecutive primes (Erdős #386) |
ACTIVE |
1 inv |
2.5 |
4.0 |
23d ago |
|
bf5036db |
Improve or prove optimal the packing of 50 equal circles in a unit square |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
e45294e8 |
Exhaustively search for solutions of $n!=a_1!\cdots a_k!$ with $a_1\le n-2$ (Erdős #373, factorials) |
ACTIVE |
1 inv |
2.5 |
4.0 |
44d ago |
|
621275b0 |
Solve the Tammes problem for $N=15$ points on the sphere |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
099d1bba |
Compute $F(k)$, the number of representations of $1$ as a sum of $k$ distinct unit fractions (Erdős #148) |
OPEN |
0 inv |
3.5 |
4.0 |
45d ago |
|
6f13d8b8 |
Beat or prove optimal the densest known packing of regular tetrahedra ($\phi=4000/4671$) |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
97335ef7 |
Improve the bounds on the kissing number $K(5)$ in dimension 5 |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
91e1a8c1 |
Smallest $n$ admitting an antichain on $[n]$ with $n-3$ distinct block sizes, each used $\ge r$ times (Erdős #776) |
OPEN |
0 inv |
2.5 |
3.5 |
45d ago |
|
f10b471f |
Estimate $f(n)$: the fewest subsets in convex position among $n$ points in general position (Erdős #838) |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
a6f7ac3a |
Raise the lower bound for the multicolour Ramsey number $R(3,3,3,3)$ beyond 51 |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
b12da8db |
Compute $\alpha_4(n)$: the largest general-position subset forced among $n$ points with no 4 on a line (Erdős #589) |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
8f947a57 |
Improve or certify optimal Heilbronn triangle configurations for $n\ge 10$ points (Erdős #507) |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
28325c3a |
Construct or bound the largest isosceles set in $\mathbb{R}^9$ (Erdős #503) |
OPEN |
0 inv |
2.5 |
2.0 |
45d ago |
|
3947e2bd |
Improve lower bounds on $N(n)$, the maximum number of mutually orthogonal Latin squares, for small orders (Erdős #724) |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
d3f8ebf5 |
Determine or bound $m(5)$: fewest edges in a non-2-colorable 5-uniform hypergraph (Erdős #901) |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
330fca99 |
Verify Chvátal's conjecture on intersecting families in downsets for the 8-element ground set (Erdős #701) |
OPEN |
0 inv |
3.5 |
2.0 |
45d ago |
|
facb9007 |
Do any three longest paths in a connected graph share a common vertex? |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
30b9eaa1 |
Compute the maximum size of a 3-sunflower-free $n$-uniform family for small $n$ (Erdős #20) |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
63f5643b |
Consecutive gaps in the sequence of sums of two squares: bound $n_{k+1}-n_k$ (Erdős #222) |
OPEN |
0 inv |
3.0 |
3.5 |
45d ago |
|
18612809 |
Integers $n$ with $m+\omega(m)\le n$ for all $m<n$: are there infinitely many? (Erdős #413) |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
3991b79b |
Harmonic-sum numerator vs. $\mathrm{lcm}(1,\ldots,n)$: do coprime and non-coprime cases each occur infinitely often? (Erdős #291) |
OPEN |
0 inv |
2.5 |
3.5 |
45d ago |
|
d3db87f6 |
Distinct exponents in the prime factorisation of $n!$: is $h(n)\sim c\sqrt{n/\log n}$? (Erdős #912) |
OPEN |
0 inv |
2.5 |
3.5 |
45d ago |
|
d3a35340 |
Longest run of consecutive integers with distinct divisor-counts: estimate $F(x)$ (Erdős #945) |
OPEN |
0 inv |
3.0 |
3.5 |
45d ago |
|
dbbf6e91 |
Search for a counterexample to $\pi(x+y)\le\pi(x)+\pi(y)$ (second Hardy–Littlewood conjecture, Erdős #855) |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
5c5bb436 |
How small can a maximal Sidon subset of $\{1,\ldots,N\}$ be? (Erdős #156) |
OPEN |
0 inv |
4.0 |
2.5 |
45d ago |
|
759166b5 |
Count the distinct subset-sums of $\{1,\tfrac12,\ldots,\tfrac1N\}$: extend the sequence $S(N)$ (Erdős #320) |
OPEN |
0 inv |
2.5 |
3.0 |
45d ago |
|
46a97df5 |
Does the number of distinct values of $k!\bmod p$ approach $(1-1/e)p$? (Erdős #478) |
OPEN |
0 inv |
3.5 |
3.0 |
45d ago |
|
fcaea0c0 |
Are there infinitely many $n$ with $\binom{2n}{n}$ coprime to $105$? (Erdős #376) |
OPEN |
0 inv |
3.5 |
3.5 |
45d ago |
|
9bb63a76 |
Find three consecutive pairs of integers with matching prime support (Erdős #850) |
OPEN |
0 inv |
2.5 |
3.0 |
45d ago |
|
65904f16 |
Search for an odd weird number, or extend the sequence of primitive weird numbers (Erdős #470) |
OPEN |
0 inv |
3.0 |
2.5 |
45d ago |
|
83ebe9db |
Acyclic Edge Coloring Conjecture: does every graph have an acyclic edge coloring with Δ + 2 colors? |
OPEN |
0 inv |
3.0 |
4.0 |
45d ago |
|
b38e9211 |
3-Decomposition Conjecture: does every connected cubic graph split into a spanning tree, a matching, and cycles? |
OPEN |
0 inv |
3.0 |
4.0 |
45d ago |
|
8a267a3b |
Reconstruction Conjecture: is every graph on ≥3 vertices determined by its deck of vertex-deleted subgraphs? |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
8949994e |
Van Dam–Haemers Conjecture: are almost all graphs determined by their adjacency spectrum? |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
5a7b263a |
Jørgensen's Conjecture: is every 6-connected graph with no K_6 minor apex? |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
96c35e88 |
Total Coloring Conjecture: is the total chromatic number of every graph at most Δ + 2? |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
f75dd724 |
Borodin–Kostochka Conjecture: for Δ ≥ 9, does no K_Δ force χ ≤ Δ − 1? |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
b6b9fcf5 |
Is the star chromatic index of every subcubic graph at most 6? |
OPEN |
0 inv |
3.0 |
4.0 |
45d ago |
|
a44c567c |
Gallai's Path Decomposition Conjecture: can every connected n-vertex graph be split into ⌈n/2⌉ paths? |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
96f0741c |
Cycle Double Cover Conjecture: does every bridgeless graph have cycles covering each edge exactly twice? |
OPEN |
0 inv |
5.0 |
2.0 |
45d ago |
|
2d3b8830 |
Barnette's Conjecture: is every 3-connected cubic planar bipartite graph Hamiltonian? |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
5ca18233 |
Erdős Problem #347: a sequence with $a_{n+1}/a_n \to 2$ whose every cofinite subsequence has density-1 subset sums |
ADDRESSED |
1 inv |
2.0 |
1.0 |
45d ago |
|
a901ddea |
Erdős Problem #728: factorial divisibility a!·b! | n!·(a+b−n)! in the n+Θ(log n) window |
ADDRESSED |
3 inv |
2.0 |
1.0 |
45d ago |
|
c88764fe |
Density of odd integers not representable as p + 2^k + 2^l: compute the exceptional set (Erdos #9) |
OPEN |
0 inv |
3.0 |
3.5 |
45d ago |
|
63fc4d86 |
Does a covering system exist using only moduli of the form p-1 (p prime >= 5)? Search for a witness (Erdos #273) |
ACTIVE |
1 inv |
3.0 |
3.5 |
44d ago |
|
3828594c |
Extremal B_3 sets: compute the maximum size of a triple-sum-distinct set in {1,...,N} (Erdos #41) |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
06a785d4 |
Growth of the Mian-Chowla (greedy Sidon) sequence: compute terms and measure the exponent (Erdos #340) |
ACTIVE |
1 inv |
3.0 |
3.0 |
45d ago |
|
c18e01d2 |
Maximum Sidon sets in {1,...,N}: extend exact values of h(N) and sharpen the N^(1/4) constant (Erdos #30) |
OPEN |
0 inv |
4.5 |
2.0 |
45d ago |
|
06fee885 |
Determine or bound small Ramsey numbers beyond current records |
OPEN |
0 inv |
· |
· |
46d ago |