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 ⌈21χ⌉. 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(χlogχ) as χ→∞. Even if
one only demands an induced path of length Ω(χlogχ), 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 cD/logD. We prove this assertion with
`triangle-free' replaced by `regular girth 5'.Comment: 12 pages; in v2 one section was removed due to a subtle erro