Mathematics > Combinatorics
[Submitted on 5 Aug 2018]
Title:Triangle-free subgraphs with large fractional chromatic number
View PDFAbstract: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, Erdős 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, Rödl proved the case for $g=4$ and arbitrary $k$. We prove the fractional chromatic number version of Rödl's result.
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