|
0e166bc5 |
Improve the largest known Condorcet domain for some n >= 9 |
OPEN |
0 inv |
3.0 |
2.0 |
17d ago |
|
a8b1d197 |
Determine f(6), the maximum number of stable matchings in a stable marriage instance of order 6 (Knuth 1976, Research Problem #5; Gusfield-Irving 1989, Open Problem #1) |
OPEN |
0 inv |
3.0 |
3.0 |
17d 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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
7d0410c7 |
How long can the primitive-set saturation game be forced to last? (Erdős #872) |
OPEN |
0 inv |
3.0 |
2.0 |
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 |
|
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 |
|
cb9bf76e |
Longest run of distinct consecutive prime gaps: estimate $h(x)$ (Erdős #852) |
OPEN |
0 inv |
3.0 |
3.5 |
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 |
|
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 |
|
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 |
|
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 |
|
f68cbd7e |
Largest subset of $\{1,\ldots,N\}$ whose pairwise sums are all squarefree (Erdős #1109) |
OPEN |
0 inv |
3.0 |
3.0 |
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 |
|
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 |
|
e8343875 |
A prime primitive root below every prime: does one always exist? (Erdős #985) |
OPEN |
0 inv |
2.5 |
3.5 |
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 |
|
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 |
|
96ee4052 |
Largest guaranteed dissociated subset f(n): is f(n) ≥ ⌊log₂ n⌋? (Erdős #963) |
ACTIVE |
2 inv |
3.0 |
2.0 |
15d 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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
7e911005 |
How large can gaps between consecutive squarefree numbers be? (Erdős #208) |
OPEN |
0 inv |
3.0 |
2.0 |
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 |
|
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 |
|
4965cda5 |
An infinite set of totient values whose smallest preimages grow superlinearly? (Erdős #51) |
ACTIVE |
1 inv |
2.0 |
2.5 |
15d 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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
fbd34fd1 |
Determine the multicolour Ramsey number $R_k(C_{2n})$ of even cycles (Erdős #555) |
OPEN |
0 inv |
3.0 |
2.5 |
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
b12dc3d6 |
Construct pairwise balanced designs with $O(\sqrt{n})$ blocks of every size (Erdős #734) |
OPEN |
0 inv |
2.0 |
2.0 |
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 |
|
cc16d0bd |
Integer-distance graphs in general position: can the chromatic number be infinite? (Erdős #130) |
OPEN |
0 inv |
3.0 |
2.0 |
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
3d5f247b |
Is every multiplicity t realized by some repeated binomial coefficient? (Singmaster-type, Erdős #849) |
OPEN |
0 inv |
4.0 |
2.0 |
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
ec17c937 |
Do $k$ consecutive primes in arithmetic progression exist for every $k$? (Erdős #141) |
OPEN |
0 inv |
3.0 |
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
4a20a96d |
Consecutive zero Taylor coefficients in Laguerre–Pólya subclasses (Hayman Problem 2.74) |
OPEN |
0 inv |
3.0 |
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 |
|
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 |
|
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 |
|
1cddfc9f |
Is the group-enumeration (gnu) function surjective onto the positive integers? (Kourovka 15.99) |
OPEN |
0 inv |
2.0 |
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
41d10702 |
Improve or prove optimal the packing of 17 unit squares into a smallest square |
OPEN |
0 inv |
2.0 |
4.0 |
45d 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 |
|
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 |
|
f0904a2d |
Improve or verify the best-known [96,40] linear code over GF(2): current bounds 20 ≤ d ≤ 26 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
bc174a43 |
Improve or verify the best-known [48,24] linear code over GF(9): current bounds 16 ≤ d ≤ 22 |
OPEN |
0 inv |
3.0 |
2.0 |
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 |
|
3b1c2482 |
Improve or verify the best-known binary constant-weight code A(29,8,7): current bounds 344 ≤ A ≤ 617 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
00470f56 |
Improve or verify the best-known binary constant-weight code A(30,6,6): current bounds 1277 ≤ A ≤ 1820 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
429ec398 |
Improve or verify the best-known [44,22] linear code over GF(4): current bounds 14 ≤ d ≤ 16 |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
fde8b707 |
Improve or verify the best-known [40,20] linear code over GF(8): current bounds 13 ≤ d ≤ 18 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
91ef483a |
Improve or verify the best-known [48,24] linear code over GF(5): current bounds 15 ≤ d ≤ 20 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
ef6668dc |
Improve or verify the best-known [60,30] linear code over GF(4): current bounds 17 ≤ d ≤ 23 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
76d3c274 |
Improve or verify the best-known [80,40] linear code over GF(3): current bounds 19 ≤ d ≤ 26 |
OPEN |
0 inv |
3.0 |
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 |
|
5d1354f0 |
Improve or verify the best-known [128,64] binary linear code: current bounds 22 ≤ d ≤ 28 |
OPEN |
0 inv |
3.0 |
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 |
|
716dd457 |
Find a synchronizing automaton with reset threshold exceeding (n-1)^2, or extend Cerny verification to n=13 |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
9a466da8 |
Improve the bounds on the football-pool number K_3(6): ternary covering code of length 6, radius 1 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
3314beef |
Is 47 multiplications optimal for 4x4 matrix multiplication over GF(2)? Beat AlphaTensor's rank-47 scheme |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
8996def9 |
Reduce the rank of the 3x3 matrix multiplication tensor below 23 (or improve the lower bound above 19) |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
49dff27d |
Improve the best-known longest coil (coil-in-the-box) in the 9-dimensional hypercube beyond length 188 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
78e7d60c |
Improve the best-known longest snake (snake-in-the-box) in the 9-dimensional hypercube beyond length 190 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
3f19c4cf |
Close the gap for the minimal superpermutation length on 6 symbols: 867 <= s(6) <= 872 |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
3a9219bd |
Improve the best-known size (comparator count) of a sorting network on 13 inputs below 45 |
OPEN |
0 inv |
3.0 |
2.0 |
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 |
|
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 |
|
f75dd724 |
Borodin–Kostochka Conjecture: for Δ ≥ 9, does no K_Δ force χ ≤ Δ − 1? |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
a9a9ed38 |
Find a smaller orphan (Garden-of-Eden pattern) in Conway's Game of Life |
OPEN |
0 inv |
2.0 |
3.0 |
45d 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 |
|
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 |