A note on multicolor Ramsey number of small odd cycles versus a large clique

Abstract

Let Rk(H;Km) be the smallest number N such that every coloring of the edges of KN with k+1 colors has either a monochromatic H in color i for some 1⩽i⩽k, or a monochromatic Km in color k+1. In this short note, we study the lower bound for Rk(H;Km) when H is C5 or C7, respectively. We show thatRk(C5;Km)=Ω(m3k8+1/(log⁡m)3k8+1), andRk(C7;Km)=Ω(m2k9+1/(log⁡m)2k9+1), for fixed positive integer k and m→∞. The proof is based on random block constructions of Mubayi and Verstraëte, who obtained comparable bounds when k=1, and random blowups argument. Our results slightly improve the previously known lower bound Rk(C2ℓ+1;Km)=Ω(mk2ℓ−1+1/(log⁡m)k+2k2ℓ−1) obtained by Alon and Rödl.