# Gallai-Ramsey number for $K_{5}$

## TL;DR

This paper investigates the Gallai-Ramsey number for $K_5$, revealing its dependence on the unknown value of the Ramsey number $R(5,5)$, and provides bounds based on different possible values of $R(5,5)$.

## Contribution

The paper advances understanding of Gallai-Ramsey numbers for $K_5$ by linking their determination to the unresolved $R(5,5)$, and offers bounds under different conjectures.

## Key findings

- If $R(5,5)
eq 43$, the conjecture is proven true.
- If $R(5,5) = 43$, bounds for $gr_k(K_3 : K_5)$ are established.
- The validity of the conjecture hinges on the actual value of $R(5,5)$.

## Abstract

Given a graph $H$, the $k$-colored Gallai Ramsey number $gr_{k}(K_{3} : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a monochromatic copy of $H$. Fox et al. [J. Fox, A. Grinshpun, and J. Pach. The Erd{\H o}s-Hajnal conjecture for rainbow triangles. J. Combin. Theory Ser. B, 111:75-125, 2015.] conjectured the value of the Gallai Ramsey numbers for complete graphs. Recently, this conjecture has been verified for the first open case, when $H = K_{4}$. In this paper we attack the next case, when $H = K_5$. Surprisingly it turns out, that the validity of the conjecture depends upon the (yet unknown) value of the Ramsey number $R(5,5)$. It is known that $43 \leq R(5,5) \leq 48$ and conjectured that $R(5,5)=43$ [B.D. McKay and S.P. Radziszowski. Subgraph counting identities and Ramsey numbers. J. Combin. Theory Ser. B, 69:193-209, 1997]. If $44 \leq R(5,5) \leq 48$, then Fox et al.'s conjecture is true and we present a complete proof. If, however, $R(5,5)=43$, then Fox et al.'s conjecture is false, meaning that at least one of these two conjectures must be false. For the case when $R(5, 5) = 43$, we show lower and upper bounds for the Gallai Ramsey number $gr_{k}(K_{3} : K_5)$.