# Separating Tree-chromatic number from Path-chromatic Number

## TL;DR

This paper demonstrates, using Ramsey theory, that some graphs can have a low tree-chromatic number while their path-chromatic number is unbounded, answering a question posed by Seymour.

## Contribution

It introduces a family of graphs with bounded tree-chromatic number but unbounded path-chromatic number, resolving a previously open problem.

## Key findings

- Tree-chromatic number can be bounded independently of path-chromatic number.
- Existence of graphs with low tree-chromatic but unbounded path-chromatic number.
- Application of Ramsey theoretic tools to graph coloring problems.

## Abstract

We apply Ramsey theoretic tools to show that there is a family of graphs which have tree-chromatic number at most~$2$ while the path-chromatic number is unbounded. This resolves a problem posed by Seymour.