|
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 |