Multicolor Ramsey numbers via pseudorandom graphs
Xiaoyu He, Yuval Wigderson

TL;DR
This paper establishes bounds on multicolor Ramsey numbers using pseudorandom graphs that are weakly optimal and $K_s$-free, extending previous results by considering a broader class of graphs.
Contribution
It generalizes existing bounds on multicolor Ramsey numbers by employing weakly optimal pseudorandom graphs, broadening the scope beyond optimal graphs.
Findings
Bounds on $r(s_1,...,s_k,t)$ are established with explicit asymptotic formulas.
Extends previous results to cases with multiple colors and larger clique sizes.
Utilizes pseudorandom graphs to improve understanding of multicolor Ramsey numbers.
Abstract
A weakly optimal -free -graph is a -regular -free graph on vertices with and spectral expansion , for some fixed . Such a graph is called optimal if additionally . We prove that if are fixed positive integers and weakly optimal -free pseudorandom graphs exist for each , 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 , where . This generalizes previous results of Mubayi and Verstra\"ete, who proved the case , and Alon and R\"odl, who proved the case . Both previous results used the existence of optimal rather than weakly optimal…
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