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

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 6 i 6 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 that Rk(C5;Km) = Ω(m 3k 8 /(logm) 3k 8 ), and Rk(C7;Km) = Ω(m 2k 9 /(logm) 2k 9 ), for fixed positive integer k and m → ∞. These slightly improve the previously known lower bound Rk(C2l+1;Km) = Ω(m k 2l−1 /(logm) 2k 2l−1 ) obtained by Alon and Rödl. The proof is based on random block constructions and random blowups argument.

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