Degree Ramsey numbers for even cycles

Abstract

Let $H\stackrel{s}{\to }G$ denote that any $s$-coloring of $E(H)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_{\Delta }(G,s)$, is $\min \{\Delta (H):H\stackrel{s}{\to }G\}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_{\Delta }(C_{2k},s)\ge 2s$, and Kang and Perarnau showed that $R_{\Delta }(C_4,s)=\Theta (s^2)$. Our main result is that $R_{\Delta }(C_6,s)=\Theta (s^{3/2})$ and $R_{\Delta }(C_{10},s)=\Theta (s^{5/4})$. Additionally, we substantially improve the lower bound for $R_{\Delta }(C_{2k},s)$ for general $k$.