Lower Bounds for Multicolor Star-Critical Ramsey Numbers

Abstract

The star-critical Ramsey number is a refinement of the concept of a Ramsey number. In this paper, we give equivalent criteria for which the star-critical Ramsey number vanishes. Next, we provide a new general lower bound for multicolor star-critical Ramsey numbers whenever it does not vanish. As an application, we evaluate $r_{\ast }(P_k,P_3,P_3)$, where $P_n$ is a path of order $n$. In the process of proving these results, we also show that $r_{\ast }(C_5,P_3)=3$, where $C_5$ is a cycle of order $5$.

18 pages, 6 figures