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

Zhuo Wu; Yisai Xue

Paper

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

Abstract

The chromatic threshold

of a graph

is the infimum of

such that the chromatic number of every

-vertex

-free graph with minimum degree at least

is bounded in terms of

and

. A breakthrough result of Allen, Böttcher, Griffiths, Kohayakawa, and Morris determined

for every graph

; in particular, if

, then

. 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

with

, allowing a constant deficit below

, we prove sharp (up to lower-order terms) upper bounds on the edge density of

-vertex

-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

; more specifically, we obtain explicit bounds in two of the three possible cases arising in the trichotomy of

. Our extremal constructions -- based on Erdős graphs and blowups of Borsuk--Hajnal graphs -- show that these bounds are best possible up to

terms.