# Induced subgraphs of graphs with large chromatic number. IX. Rainbow paths

## TL;DR

This paper proves that graphs with bounded clique number and sufficiently large chromatic number contain rainbow paths of any specified length, regardless of vertex coloring, extending previous results for special cases.

## Contribution

It generalizes earlier work by showing the existence of rainbow paths in graphs with arbitrary clique number and large chromatic number, independent of coloring optimality.

## Key findings

- Existence of rainbow paths of any length in such graphs
- Extension of previous results from girth constraints to clique number constraints
- Applicable to all nonnegative integers k,s

## Abstract

We prove that for all nonnegative integers k,s there exists c with the following property. Let G be a graph with clique number at most k and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. This extends a recent result of Gyarfas and Sarkozy, who proved the same (when k=2) for graphs G with girth at least five.