Interpolating chromatic and homomorphism thresholds

TL;DR

This work develops a VC-dimension–driven framework to unify and interpolate chromatic and homomorphism thresholds for $H$-free graphs, introducing a precise formula $oldsymbol{ au}_ ext{hom}^{ ext{VC}}(K_s;K_t)=\frac{(s-3)(t-s+2)+1}{(s-2)(t-s+2)+1}$ that tracks how restricting homomorphic images affects density thresholds. It also defines the blowup threshold $oldsymbol{ au}_B(H)$ and proves $oldsymbol{ au}_B(C_{2k-1})=1/(2k-1)$, establishing that 0 is an accumulation point for blowup thresholds and highlighting a separation from chromatic thresholds. The proofs weave VC-dimension packing, iterative partition refinements, and coding-on-graphs concepts, yielding both lower-bound constructions and upper-bound refinements that culminate in bounded-size homomorphic images for dense, $H$-free graphs. Collectively, the results unify prior work of Thomassen, Łuczak–Thomassé, Goddard–Lyle–Nikiforov, and Schacht, while providing algorithmic, combinatorial, and structural insights into how high-density, forbidden-subgraph settings shape permissible homomorphic images and blowups.

Abstract

The problem of chromatic thresholds seeks for minimum degree conditions that ensure $H$-free graphs to have a bounded chromatic number, or equivalently a bounded size homomorphic image. The strengthened homomorphism thresholds problem further requires that the homomorphic image itself is $H$-free. The purpose of this paper is two-fold. First, we define a generalized notion of threshold which encapsulates and interpolates chromatic and homomorphism thresholds via the theory of VC-dimension. Our first result shows a smooth transition between these two thresholds when varying the restrictions on homomorphic images. In particular, we proved that for $t \ge s \ge 3$ and $ε>0$, if $G$ is an $n$-vertex $K_s$-free graph with VC-dimension $d$ and $δ(G) \ge (\frac{(s-3)(t-s+2)+1}{(s-2)(t-s+2)+1} + ε)n$, then $G$ is homomorphic to a $K_t$-free graph $H$ with $|H| = O(1)$. Moreover, we construct graphs showing that this minimum degree condition is optimal. This extends and unifies the results of Thomassen, Łuczak and Thomassé, and Goddard, Lyle and Nikiforov, and provides a deeper insight into the cause of existences of homomorphic images with various properties. Second, we introduce the blowup threshold $δ_B(H)$ as the infimum $α$ such that every $n$-vertex maximal $H$-free graph $G$ with $δ(G)\geαn$ is a blowup of some $F$ with $|F|=O(1)$. This notion strengthens homomorphism threshold. While the homomorphism thresholds for odd cycles remain unknown, we prove that $δ_B(C_{2k-1})=1/(2k-1)$ for any integer $k\ge 2$. This strengthens the result of Ebsen and Schacht and answers a question of Schacht and shows that, in sharp contrast to the chromatic thresholds, 0 is an accumulation point for blowup thresholds. Our proofs mix tools from VC-dimension theory and an iterative refining process, and draw connection to a problem concerning codes on graphs.

Figures (3)

• Figure 1.1: The $x,y,z$-axes correspond to the values of $t-s$, $s$ and $\delta_{\textup{hom}}^{\textup{VC}}(K_s;K_t)$, respectively. The curve on the plane $x=0$ corresponds to $\delta_{\textup{hom}}^{\textup{VC}}(K_s)$; the curves on the planes $y=s$ correspond to $\delta_{\textup{hom}}^{\textup{VC}}(K_s;K_t)$ respectively and their limits when $t\to \infty$ form the curve $\delta_{\chi}^{\textup{VC}}(K_s)$ on the plane $x=+\infty$.
• Figure 2.1: Structure of the 6-partite graph $G$ with $t=6$, $m=2$
• Figure 2.2: Structure of the 5-partite graph $G$ with $t=5$, $m=4$

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Definition 2.2
• Claim 2.3
• proof : Proof of claim
• Definition 2.4