Multicolor Ramsey numbers via pseudorandom graphs

A weakly optimal Ks-free (n,d,λ)-graph is a d-regular Ks-free graph on n vertices with d=Θ(n1−α) and spectral expansion λ=Θ(n1−(s−1)α), for some fixed α>0. Such a graph is called optimal if additionally α=12s−3. We prove that if s1,…,sk≥3 are fixed positive integers and weakly optimal Ksi-free pseudorandom graphs exist for each 1≤i≤k, then the multicolor Ramsey numbers satisfy Ω(tS+1log2St)≤r(s1,…,sk,t)≤O(tS+1logSt), as t→∞, where S=∑ki=1(si−2). This generalizes previous results of Mubayi and Verstraëte, who proved the case k=1, and Alon and Rödl, who proved the case s1=⋯=sk=3. Both previous results used the existence of optimal rather than weakly optimal Ksi-free graphs.