Ramsey Numbers of Large Even Cycles and Fans

Abstract

For graphs $F$ and $H$, the Ramsey number $R(F, H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red $F$ or a blue $H$. Let $C_n$ be a cycle of length $n$ and $F_n$ be a fan consisting of $n$ triangles all sharing a common vertex.In this paper, we prove that for all sufficiently large $n$,\[R(C_{2\lfloor an\rfloor}, F_n)= \left\{ \begin{array}{ll}(2+2a+o(1))n & \textrm{if $1/2\leq a< 1$,}\\(4a+o(1))n & \textrm{if $ a\geq 1$.}\end{array} \right.\]