Factorization norms and Zarankiewicz problems

TL;DR

This paper explores the combinatorial properties of the $oldsymbol{_2}$-norm of Boolean matrices, establishing bounds on the number of ones in matrices with certain sparsity and norm constraints, with applications to Zarankiewicz problems and geometric graph theory.

Contribution

It introduces new bounds linking the $oldsymbol{
3_2}$-norm to sparsity and degree bounds, addressing conjectures and extending results in geometric and combinatorial graph theory.

Findings

Bound on the number of ones in matrices with bounded $oldsymbol{
3_2}$-norm and no large all-ones submatrix

Average degree bounds for $K_{t,t}$-free incidence graphs of points and polytopes

Extension of results to semilinear graphs, strengthening prior work

Abstract

The $\gamma_2$-norm of Boolean matrices plays an important role in communication complexity and discrepancy theory. In this paper, we study combinatorial properties of this norm, and provide new applications, involving Zarankiewicz type problems. We show that if $M$ is an $m\times n$ Boolean matrix such that $\gamma_2(M)<\gamma$ and $M$ contains no $t\times t$ all-ones submatrix, then $M$ contains $O_{\gamma,t}(m+n)$ one entries. In other words, graphs of bounded $\gamma_2$-norm are degree bounded. This addresses a conjecture of Hambardzumyan, Hatami, and Hatami for locally sparse matrices. We prove that if $G$ is a $K_{t,t}$-free incidence graph of $n$ points and $n$ homothets of a polytope $P$ in $\mathbb{R}^d$, then the average degree of $G$ is $O_{d,P}(t(\log n)^{O(d)})$. This is sharp up the $O(.)$ notations. In particular, we prove a more general result on semilinear graphs, which greatly strengthens the work of Basit, Chernikov, Starchenko, Tao, and Tran.