Multicolor Ramsey Numbers via Pseudorandom Graphs

Abstract

A weakly optimal $K_s$-free $(n,d,\lambda)$-graph is a $d$-regular $K_s$-free graph on $n$ vertices with $d=\Theta(n^{1-\alpha})$ and spectral expansion $\lambda=\Theta(n^{1-(s-1)\alpha})$, for some fixed $\alpha>0$. Such a graph is called optimal if additionally $\alpha = \frac{1}{2s-3}$. We prove that if $s_{1},\ldots,s_{k}\ge3$ are fixed positive integers and weakly optimal $K_{s_{i}}$-free pseudorandom graphs exist for each $1\le i\le k$, then the multicolor Ramsey numbers satisfy\[\Omega\Big(\frac{t^{S+1}}{\log^{2S}t}\Big)\le r(s_{1},\ldots,s_{k},t)\le O\Big(\frac{t^{S+1}}{\log^{S}t}\Big),\]as $t\rightarrow\infty$, where $S=\sum_{i=1}^{k}(s_{i}-2)$. This generalizes previous results of Mubayi and Verstra\"ete, who proved the case $k=1$, and Alon and Rödl, who proved the case $s_1=\cdots = s_k = 3$. Both previous results used the existence of optimal rather than weakly optimal $K_{s_i}$-free graphs.

Making an $H$-Free Graph $k$-Colorable

“…However, it is conjectured that such a graph exists. If so, the proof would carry over and we would obtain the following conjecture, which would show that Theorem 1.1 is tight up to a constant function of r for sufficiently large n. It is known that other interesting results would follow from knowing the existence of such pseudorandom K r -free graphs, including giving nearly tight bounds for off-diagonal Ramsey numbers (see [33,38]).…”

Making an H $H$‐free graph k $k$‐colorable

“…It is known that other interesting results would follow from knowing the existence of such pseudorandom $${K}_{r}$$‐free graphs, including giving nearly tight bounds for off‐diagonal Ramsey numbers (see [26, 30]).…”

“…However, it is conjectured that such a graph exists. If so, the proof would carry over and we would obtain the following conjecture, which would show that Theorem 1.1 is tight up to a constant function of r for sufficiently large n. It is known that other interesting results would follow from knowing the existence of such pseudorandom K r -free graphs, including giving nearly tight bounds for off-diagonal Ramsey numbers (see [26,30]).…”

A clique-free pseudorandom subgraph of the pseudo polarity graph

“…It is known (using the expander mixing lemma, see for example [AS16, Corollary 9.2.5]) that the edge density of an optimally pseudorandom K k -free graph is O(n −1/(2k−3) ). If optimally pseudorandom graphs with this edge density exist, it follows from the results of Mubayi and Verstraëte [MV19], and the multicolour generalization by He and Wigderson [HW20], that the determination of the off-diagonal Ramsey number r(k 1 , . .…”