Computing the Ramsey Number $R(K_5-P_3,K_5)$

TL;DR

This paper uses computer-assisted methods to determine the exact Ramsey number R(K_5-P_3, K_5)=25, resolving a long-standing open case and identifying the unique good graph structures on fewer vertices.

Contribution

It provides the first computer-assisted proof of this Ramsey number and characterizes the structure of all relevant graphs on fewer vertices.

Findings

No (K_5-P_3,K_5)-good graphs with a K_4 on 23 or 24 vertices.

The unique (K_5-P_3,K_5)-good graph with a K_4 on 22 vertices.

Confirmation that R(K_5-P_3,K_5)=25.

Abstract

We give a computer-assisted proof of the fact that $R(K_5-P_3, K_5)=25$. This solves one of the three remaining open cases in Hendry's table, which listed the Ramsey numbers for pairs of graphs on 5 vertices. We find that there exist no $(K_5-P_3,K_5)$-good graphs containing a $K_4$ on 23 or 24 vertices, where a graph $F$ is $(G,H)$-good if $F$ does not contain $G$ and the complement of $F$ does not contain $H$. The unique $(K_5-P_3,K_5)$-good graph containing a $K_4$ on 22 vertices is presented.