# Gallai-Ramsey numbers for rainbow paths

## TL;DR

This paper investigates Gallai-Ramsey numbers for rainbow paths, providing complete solutions for P4 and conjectures supported by results for P5, linking these numbers to classical Ramsey numbers.

## Contribution

The paper fully solves the Gallai-Ramsey problem for P4 and proposes a conjecture for P5, connecting these numbers to known Ramsey numbers.

## Key findings

- Complete solution for P4 case reducing to 2-color Ramsey numbers.
- Conjecture that P5 case reduces to 3-color Ramsey numbers.
- Supporting results for the P5 conjecture.

## Abstract

Given graphs $G$ and $H$ and a positive integer $k$, the \emph{Gallai-Ramsey number}, denoted by $gr_{k}(G : H)$ is defined to be the minimum integer $n$ such that every coloring of $K_{n}$ using at most $k$ colors will contain either a rainbow copy of $G$ or a monochromatic copy of $H$. We consider this question in the cases where $G \in \{P_{4}, P_{5}\}$. In the case where $G = P_{4}$, we completely solve the Gallai-Ramsey question by reducing to the $2$-color Ramsey numbers. In the case where $G = P_{5}$, we conjecture that the problem reduces to the $3$-color Ramsey numbers and provide several results in support of this conjecture.