Degree bipartite Ramsey numbers

Abstract

Let $H\stackrel{s}{\to }G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_{\Delta }(G;s)$ is defined to be $\min \{\Delta (H):H\stackrel{s}{\to }G\}$, and the degree bipartite Ramsey number $b{r}_{\Delta }(G;s)$ is defined to be $\min\{Δ(H):H\xrightarrow{s} G\; \mbox{and} \;χ(H)=2\}$. In this note, we show that $r_{\Delta }(K_{m,n};s)$ is linear on $n$ with $m$ fixed. We also determine $b{r}_{\Delta }(G;s)$ where $G$ are trees, including stars and paths, and complete bipartite graphs.

7 pages