Abstract
Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\rceil$. 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 $\chi$ contains an induced cycle of length $\Omega(\chi\log\chi)$ as $\chi\to\infty$. Even if one only demands an induced path of length $\Omega(\chi\log\chi)$, 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'.
Highlights
For graphs with bounded clique number ω, the tradeoff between chromatic number χ being high and there being certain induced subgraphs is a central topic in graph theory
This already has some history in the area: for instance, Kierstead and Trotter [16] pursued this in attempts towards the Gyarfas– Sumner Conjecture
Our starting point is a more explicit form of this tradeoff, where the commodities are instead proper colourings and rainbow induced subgraphs. It is interesting in its own right and some recent activity [4, 10, 21] has been motivated by a conjecture of this form due to the first author
Summary
For graphs with bounded clique number ω, the tradeoff between chromatic number χ being high and there being certain induced subgraphs is a central topic in graph theory. Every properly coloured triangle-free graph of chromatic number χ contains a rainbow induced path of length χ. Every Kr-free graph of chromatic number χ contains an independent set of size c2χ1/(r−2) log χ This immediately yields the following result complementing Theorem 2. For each r 4 there are c > 0 and a Kr-free graph of chromatic number χ such that no matter the proper colouring it contains no rainbow independent set of size cχ2/(r−1)(log χ)(r2−r−4)/((r−2)(r−1)). Conjecture 9 would constitute a common generalisation of Johansson’s theorem, Conjecture 6 and the fact that χ(G) = O(α(G)/ log α(G)) for every triangle-free graph G 1, corresponding to the cases where H is a star, a path and an independent set, respectively. We have proved a form of Conjecture 9 for regular girth 5 graphs; see Corollary 18
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