A note on pseudorandom Ramsey graphs

Abstract

For fixed s ≥ 3, we prove that if optimal Ks-free pseudorandom graphs exist, then the Ramsey number r.s; t / is t s-1Co.1/ as t ! 1. Our method also improves the best lower bounds for r.Cℓ; t / obtained by Bohman and Keevash from the random Cℓ-free process by polylogarithmic factors for all odd ℓ ≥ 5 and ℓ 2 16; 10o. For ℓ D 4 it matches their lower bound from the C4-free process. We also prove, via a different approach, that r.C5; t / > .1 C o.1//t11=8 and r.C7; t / > .1 C o.1//t11=9. These improve the exponent of t in the previous best results and appear to be the first examples of graphs F with cycles for which such an improvement of the exponent for r.F; t / is shown over the bounds given by the random F -free process and random graphs.