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

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 for R k (H ; K m) when H is C 5 or C 7 , respectively. We show that R k (C 5 ; K m) = Ω (m 3 k 8 + 1 / (log ⁡ m) 3 k 8 + 1) , and R k (C 7 ; K m) = Ω (m 2 k 9 + 1 / (log ⁡ m) 2 k 9 + 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 R k (C 2 ℓ + 1 ; K m) = Ω (m k 2 ℓ − 1 + 1 / (log ⁡ m) k + 2 k 2 ℓ − 1 ) obtained by Alon and Rödl. [ABSTRACT FROM AUTHOR]