A note on Pseudorandom Ramsey graphs

09/03/2019

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by   Dhruv Mubayi, et al.

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For fixed s > 3, we prove that if optimal K_s-free pseudorandom graphs exist, then the Ramsey number r(s,t) = t^s-1+o(1) as t →∞. Our method also improves the best lower bounds for r(C_ℓ,t) obtained by Bohman and Keevash from the random C_ℓ-free process by polylogarithmic factors for all odd ℓ≥ 5 and ℓ∈{6,10}. For ℓ = 4 it matches their lower bound from the C_4-free process. We also prove, via a different approach, that r(C_5, t)> (1+o(1))t^11/8 and r(C_7, t)> (1+o(1))t^11/9. 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.