Edge density and minimum degree thresholds for $H$-free graphs with unbounded chromatic number

Abstract: The chromatic threshold $δχ(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $dn$ is bounded in terms of $H$ and $d$. A breakthrough result of Allen, Böttcher, Griffiths, Kohayakawa, and Morris determined $δχ(H)$ for every graph $H$; in particular, if $χ(H)=r\ge 3$, then $δχ(H) \in{\frac{r-3}{r-2},~\frac{2 r-5}{2 r-3},~\frac{r-2}{r-1}}$. In this paper we investigate the trade-off between minimum degree and edge density in the critical window around the chromatic threshold. For a fixed graph $H$ with $χ(H)=r$, allowing a constant deficit below $δχ(H)$, we prove sharp (up to lower-order terms) upper bounds on the edge density of $n$-vertex $H$-free graphs whose chromatic number diverges. Equivalently, within this degree regime we show that a suitable global bound on the number of edges forces the chromatic number to remain bounded. Our results thus quantify how global edge density can compensate for a deficit in the local minimum-degree condition near $δχ(H)$; more specifically, we obtain explicit bounds in two of the three possible cases arising in the trichotomy of $δχ(H)$. Our extremal constructions -- based on Erdős graphs and blowups of Borsuk--Hajnal graphs -- show that these bounds are best possible up to $o(n^2)$ terms.