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arXiv · 2609.40192

On the solution to the Erdős-Hajnal problem on high-girth high-chromatic subgraphs

Tung Nguyen·Bartosz Walczak
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Abstract

A well-known problem of Erdős and Hajnal from the 1960s asks whether every graph with huge chromatic number contains a subgraph with large girth and large chromatic number. Very recently, Kohlmeyer and Kruer provided a strong negative solution to this problem: a construction of triangle-free graphs with arbitrarily large chromatic number whose subgraphs with no four-cycle have chromatic number at most $6$. The purpose of this exposition is to explain the construction method, relate it to relevant literature, and optimise the bound `$6$' to `$3$'.

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Tung Nguyen, Bartosz Walczak. 2026-09-30. On the solution to the Erdős-Hajnal problem on high-girth high-chromatic subgraphs. https://arxiv.org/abs/2609.40192

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