Exact Values of Multicolor Ramsey Numbers $$R_l(C_{\le l+1})$$

Abstract

Let $$ex(n, C_{\le m})$$ denote the maximum size of a graph of order n and girth at least $$m+1$$, and $$EX(n, C_{\le m})$$ be the set of all graphs of girth at least $$m+1$$ and size $$ex(n, C_{\le m})$$. The Ramsey number $$R_l(C_{\le m})$$ is the smallest n such that every $$K_n$$, whose edges are in l colors, must contain a monochromatic cycle of length k for some $$3\le k\le m$$. In this paper, we study the exact values of $$R_l(C_{\le l+1})$$. By using the known results of $$ex(n, C_{\le l+1})$$, we first give the upper bounds on $$R_l(C_{\le l+1})$$, then we prove that $$R_l(C_{\le l+1})=2l+3$$ for odd $$l\ge 3$$. For even l, we prove that $$R_4(C_{\le 5})=12$$, $$R_6(C_{\le 7})=16$$, and $$R_l(C_{\le l+1})=2l+3$$ for $$8\le l\le 12$$, leaving the case of $$l\ge 14$$ open.