|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
06fee885 |
Determine or bound small Ramsey numbers beyond current records |
OPEN |
0 inv |
· |
· |
46d ago |