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

Abstract

Let R k (H; K m ) be the smallest number N such that every coloring of the edges of K N with k + 1 colors has either a monochromatic H in color i for some 1 i k, or a monochromatic K m in color k + 1. In this short note, we study the lower bound for3k 8 +1 ), and R k (C 7 ; K m ) = Ω(m 2k 9 +1 /(log m) 2k 9 +1 ), for fixed positive integer k and m → ∞. These slightly improve the previously known lower bound R k (C 2ℓ+1 ; K m ) = Ω(m k 2ℓ−1 +1 /(log m) k+ 2k 2ℓ−1 ) obtained by Alon and Rödl. The proof is based on random block constructions and random blowups argument.