Induced subdivisions in $K_{s,s}$-free graphs with polynomial average degree

TL;DR

This work addresses when graphs with high average degree, excluding a fixed biclique, must contain either a $K_{s,s}$ or an induced subdivision of a fixed graph $H$, proving a conjecture of Bonamy et al. via a robust cleaning/dichotomy framework. The authors introduce 1-subdivision techniques of hypergraphs to convert density into induced subdivisions, and establish a strong polychotomy: for large density graphs one must find either a clique, an induced biclique, or an induced subdivision of a fixed graph. These results yield polynomial degree-bounding for degree-bounded hereditary classes and imply polynomial χ-boundedness in several families, resolving several questions in the area and connecting to contemporary work on induced subdivisions. The methods blend dependent random choice, subdivision reductions, and VC-dimension–style shattering to translate global density into local induced configurations, with tightness results up to constant factors. Overall, the paper advances our understanding of degree-driven structure in graphs free of $K_{s,s}$ and provides key tools for further progress on χ-boundedness and related questions in graph theory.

Abstract

In this paper we prove that for every $s\geq 2$ and every graph $H$ the following holds. Let $G$ be a graph with average degree $Ω_H(s^{C|H|^2})$, for some absolute constant $C>0$, then $G$ either contains a $K_{s,s}$ or an induced subdivision of $H$. This is essentially tight and confirms a conjecture of Bonamy, Bousquet, Pilipczuk, Rzążewski, Thomassé, and Walczak. A slightly weaker form of this has been independently proved by Bourneuf, Bucić, Cook, and Davies. We actually prove a much more general result which implies the above (with worse dependence on $|H|$). We show that for every $ k\geq 2$ there is $C_k>0$ such that any graph $G$ with average degree $s^{C_k}$ either contains a $K_{s,s}$ or an induced subgraph $G'\subseteq G$ without $C_4$'s and with average degree at least $k$. Finally, using similar methods we can prove the following. For every $k,t\geq 2$ every graph $G$ with average degree at least $C_tk^{Ω(t)}$ must contain either a $K_k$, an induced $K_{t,t}$ or an induced subdivision of $K_k$. This is again essentially tight up to the implied constants and answers in a strong form a question of Davies.

Theorems & Definitions (76)

• Theorem 1.1: Kühn and Osthus
• Theorem 1.2
• Theorem 1.3
• Remark
• Theorem 1.5
• Remark
• Remark
• Theorem 2.1
• Lemma 2.2
• proof