Year: 2025
Type: article
Abstract: In this paper we prove that for every $s\geqslant 2$ and every graph $H$ the following holds. Let $G$ be a graph with average degree $\Omega _{H}(s^{A|H|^{2}})$, for some absolute constant $A>0$, 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, Rz kaŻewski, Thomassé, and Walczak in [2]. A slightly weaker form of this has been independently proved by Bourneuf, Bucić, Cook, and Davies [3]. We actually prove a much more general result, which implies the above (with worse dependence on $|H|$). We show that for every $ k\geqslant 2$ there is $A_{k}>0$ such that any graph $G$ with average degree $s^{A_{k}}$ either contains a $K_{s,s}$ or an induced subgraph $G^{\prime}\subseteq G$ without $C_{4}$’s and with average degree at least $k$. Finally, using similar methods we can prove the following. For every $k,t\geqslant 2$ every graph $G$ with average degree at least $B_{t}k^{\Omega (t)}$ 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.
DOI: 10.1093/imrn/rnaf025
Authors: António Girão, Zach Hunter
Institutions: University of Oxford, ETH Zurich
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,534
OpenAlex ID: W4407696633