Abstract:
 We introduce the $\it circular chromatic Ramsey number$ $R_{\chi_c}({\cal F},{\cal G})$ as the infimum of the circular chromatic numbers of graphs $H$ such that every red/blue edgecoloring of $H$ yields a red copy of a graph in $\cal F$ or a blue copy of a graph in $\cal G$. We prove $R_{\chi_c}(K_3,K_3)=6$ and $R_{\chi_c}(K_3,K_4)=9$.
Also, if $\cal F$ and $\cal G$ each contain a graph with circular chromatic number at most $5/2$ (such as any odd cycle of length at least $5$), then $R_{\chi_c}({\cal F},{\cal G})=4$. Furthermore, no graph has circular chromatic Ramsey number between $4$ and $5$.
Joint work with Claude Tardif, Douglas West, and Xuding Zhu.
