Stability with minuscule structure for chromatic thresholds

TL;DR

This work advances the understanding of near-threshold structure for chromatic thresholds by proving stability theorems corresponding to the previously determined δχ(H) values. Using a combination of regularity-lemma techniques, decomposition into macro- and micro-structures, and carefully constructed extremal graphs (notably Kneser- and Erdős-type objects), the authors show that almost-extremal H-free graphs with high chromatic number must resemble the known extremal configurations, with strong micro-structure constraints in the clique case. They also extend the paradigm to fractional and bounded-VC thresholds, providing exact determinations that align with and illuminate the combinatorial mechanisms at play. The results yield precise edge-edit proximity to Turán-type graphs and, in the clique case, a polylogarithmic bound on a small exceptional component’s structure, thereby enriching the toolkit for stability in extremal graph theory and offering new avenues for further refinement and generalization.

Abstract

The chromatic threshold $δ_χ(H)$ 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 $d n$ is bounded by a constant depending only on $H$ and $d$. Allen, B{ö}ttcher, Griffiths, Kohayakawa, and Morris determined the chromatic threshold for every $H$; in particular, they showed that if $χ(H)=r\ge 3$, then $δ_χ(H) \in\{\frac{r-3}{r-2},~\frac{2 r-5}{2 r-3},~\frac{r-2}{r-1}\}$. While the chromatic thresholds have been completely determined, rather surprisingly the structural behaviors of extremal graphs near the threshold remain unexplored. In this paper, we establish the stability theorems for chromatic threshold problems. We prove that every $n$-vertex $H$-free graph $G$ with $δ(G)\ge (δ_χ(H)-o(1))n$ and $χ(G)=ω(1)$ must be structurally close to one of the extremal configurations. Furthermore, we give a stronger stability result when $H$ is a clique, showing that $G$ admits a partition into independent sets and a small subgraph on sublinear number of vertices. We show that this small subgraph has fractional chromatic number $2+o(1)$ and is homomorphic to a Kneser graph defined by subsets of a logarithmic size set; both these two bounds are best possible. This is the first stability result that captures the lower-order structural features of extremal graphs. We also study two variations of chromatic thresholds. Replacing chromatic number by its fractional counterpart, we determine the fractional chromatic thresholds for all graphs. Another variation is the bounded-VC chromatic thresholds, which was introduced by Liu, Shangguan, Skokan, and Xu very recently. Extending work of Łuczak and Thomass{é} on the triangle case, we determine the bounded-VC chromatic thresholds for all cliques.

Figures (2)

• Figure 1: The Hajnal graph $H(k, \ell, m)$.
• Figure 2: The shift graph $\mathrm{Sh}_5^2$

Theorems & Definitions (69)

• Theorem 1.1: Stability for $\delta_{\chi}(K_r)$
• Theorem 1.2: Stability for $\delta_\chi(H)=\frac{2r-5}{2r-3}$
• Theorem 1.3: Stability for $\delta_\chi(H)=\frac{r-3}{r-2}$
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: allen2013chromatic
• Definition 2.3: Hajnal graphs
• Remark 2.6
• Theorem 2.7: Theorem 1.10 of komlos1995szemeredi
• Theorem 2.8: Counting lemma, Theorem 3.1 komlos1995szemeredi