Multicolor Ramsey numbers via pseudorandom graphs

Abstract

A weakly optimal $K_s$-free $(n,d,λ)$-graph is a $d$-regular $K_s$-free graph on $n$ vertices with $d=Θ(n^{1-α})$ and spectral expansion $λ=Θ(n^{1-(s-1)α})$, for some fixed $α>0$. Such a graph is called optimal if additionally $α= \frac{1}{2s-3}$. 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 \[ Ω\Big(\frac{t^{S+1}}{\log^{2S}t}\Big)\le r(s_{1},\ldots,s_{k},t)\le O\Big(\frac{t^{S+1}}{\log^{S}t}\Big), \] as $t\rightarrow\infty$, where $S=\sum_{i=1}^{k}(s_{i}-2)$. This generalizes previous results of Mubayi and Verstraëte, 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.