Lower Bounds for Multicolor Star-Critical Ramsey Numbers

TL;DR

This paper investigates the properties of multicolor star-critical Ramsey numbers, establishing criteria for their vanishing, providing new lower bounds, and applying these results to specific path and cycle graphs.

Contribution

It introduces equivalent criteria for when star-critical Ramsey numbers vanish and offers a new general lower bound for multicolor cases, with specific evaluations for paths and cycles.

Findings

Star-critical Ramsey number vanishes under certain conditions.

New lower bounds for multicolor star-critical Ramsey numbers.

Explicit values for specific graph configurations like $r_*(P_k, P_3, P_3)$ and $r_*(C_5, P_3)$.

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 $r_*(P_k, P_3, P_3)$, where $P_n$ is a path of order $n$. In the process of proving these results, we also show that $r_*(C_5, P_3)=3$, where $C_5$ is a cycle of order $5$.