Structure and colour in triangle-free graphs

Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $χ$ contains a rainbow independent set of size $\lceil\frac12χ\rceil$. This is sharp up to a factor $2$. This result and its short proof have implications for the related notion of chromatic discrepancy. Drawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number $χ$ contains an induced cycle of length $Ω(χ\logχ)$ as $χ\to\infty$. Even if one only demands an induced path of length $Ω(χ\logχ)$, the conclusion would be sharp up to a constant multiple. We prove it for regular girth $5$ graphs and for girth $21$ graphs. As a common strengthening of the induced paths form of this conjecture and of Johansson's theorem (1996), we posit the existence of some $c >0$ such that for every forest $H$ on $D$ vertices, every triangle-free and induced $H$-free graph has chromatic number at most $c D/\log D$. We prove this assertion with `triangle-free' replaced by `regular girth $5$'.

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