Degree Ramsey numbers for cycles and blowups of trees

Abstract

Let $H\stackrel{s}{\to }G$ mean that every $s$-coloring of $E\left(H\right)$ produces a monochromatic copy of $G$ in some color class. Let the $s$-color degree Ramsey number of a graph $G$, written $R_{\Delta }(G;s)$, be $min\{\Delta \left(H\right):H\stackrel{s}{\to }G\}$. We prove that the $2$-color degree Ramsey number is at most $96$ for every even cycle and at most $3458$ for every odd cycle. For the general $s$-color problem on even cycles, we prove $R_{\Delta }(C_{2m};s)\le 16s^6$ for all $m$, and $R_{\Delta }(C_4;s)\ge 0.007s^{14/9}$. The constant upper bound for$R_{\Delta }(C_n;2)$ uses blowups of graphs, where the $d$-blowup of a graph $G$ is the graph $G^{\prime }$ obtained by replacing each vertex of $G$ with an independent set of size $d$ and each edge $e$ of $G$ with a copy of the complete bipartite graph $K_{d,d}$. We also prove the existence of a function $f$ such that if $G^{\prime }$ is the $d$-blowup of $G$, then $R_{\Delta }(G^{\prime };s)\le f(R_{\Delta }(G;s),s,d)$.