Triangle-free subgraphs with large fractional chromatic number

TL;DR

This paper proves a fractional chromatic number version of a longstanding conjecture, showing that graphs with sufficiently large fractional chromatic number contain subgraphs with high chromatic number and girth.

Contribution

It extends R"{o}dl's 1977 result to fractional chromatic numbers, advancing understanding of graph coloring properties.

Findings

Established the fractional chromatic number analogue of R"{o}dl's theorem

Demonstrated existence of subgraphs with high fractional chromatic number and girth

Contributed to the theory of fractional graph coloring

Abstract

It is well known that for any integers $k$ and $g$, there is a graph with chromatic number at least $k$ and girth at least $g$. In 1960's, Erd\H{o}s and Hajnal conjectured that for any $k$ and $g$, there exists a number $h(k,g)$, such that every graph with chromatic number at least $h(k,g)$ contains a subgraph with chromatic number at least $k$ and girth at least $g$. In 1977, R\"{o}dl proved the case for $g=4$ and arbitrary $k$. We prove the fractional chromatic number version of R\"{o}dl's result.