The Bounded-VC chromatic thresholds of graphs

Abstract

For a graph $H$, the chromatic threshold ${\delta }_{\chi }(H)$ is the infimum of $c>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $cn$ is bounded by a constant depending only on $H$ and $c$. Allen, Böttcher, Griffiths, Kohayakawa, and Morris proved that if $\chi (H)=r\ge 3$, then ${\delta }_{\chi }(H)\in \{\frac{r-3}{r-2},\frac{2r-5}{2r-3},\frac{r-2}{r-1}\}$. Liu, Shangguan, Skokan, and Xu introduced the bounded-VC chromatic threshold $\text{VC}(H)$ by restricting the host graphs to have bounded VC-dimension. We determine this parameter for graph $H$ with $\chi (H)\ge 3$. More precisely, let $\mathcal{M}(H)$ be the decomposition family of an $r$-chromatic graph $H$, then \[ \text{VC}(H)= \begin{cases} \dfrac{r-3}{r-2},&\text{if $\mathcal{M}(H)$ contains a forest},\\[4pt] \dfrac{r-2}{r-1},&\text{otherwise}. \end{cases} \]