$C_4$-free subgraphs of high degree with geometric applications

TL;DR

This paper introduces a new structural approach to Zarankiewicz problems, particularly in geometric graph classes, providing optimal bounds for $C_4$-free subgraphs with high degree and their applications.

Contribution

It develops a novel structural tool that either finds a high-degree $C_4$-free subgraph or a dense subgraph, advancing solutions to geometric Zarankiewicz problems.

Findings

Established a dichotomy for graphs with given average degree

Derived optimal bounds for geometric Zarankiewicz problems

Unified approach applicable to various geometric graph classes

Abstract

The Zarankiewicz problem, a cornerstone problem in extremal graph theory, asks for the maximum number of edges in an $n$-vertex graph that does not contain the complete bipartite graph $K_{s,s}$. While the problem remains widely open in the case of general graphs, the past two decades have seen significant progress on this problem for various restricted graph classes -- particularly those arising from geometric settings -- leading to a deeper understanding of their structure. In this paper, we develop a new structural tool for addressing Zarankiewicz-type problems. More specifically, we show that for any positive integer $k$, every graph with average degree $d$ either contains an induced $C_4$-free subgraph with average degree at least $k$, or it contains a $d$-vertex subgraph with $\Omega_k(d^2)$ edges. As an application of this dichotomy, we propose a unified approach to a large number of Zarankiewicz-type problems in geometry, obtaining optimal bounds in each case.