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Factorization norms and Zarankiewicz problems

István Tomon

TL;DR

This work studies the γ2-norm (max-norm) of Boolean matrices, linking spectral decompositions to combinatorial structure. It establishes that four-cycle-free Boolean matrices with degeneracy d satisfy γ2(M) = Θ(√d), enabling degree-boundedness results: matrices with bounded γ2-norm and no large all-ones submatrices have few ones, and various decompositions (into thin blocky matrices) are equivalent to bounded degeneracy. These insights yield Zarankiewicz-type incidence bounds for point–polytope configurations, showing that K_{t,t}-free incidence graphs between n points and polytopes from POL(H) have edges bounded by O_{s,t}(n (log n)^{O(d)}) in semilinear settings, with the log-exponent depending only on dimension. The paper also connects to discrepancy theory via the Matoušek–Nikolov–Talwar inequality, demonstrating that γ2 controls hereditary discrepancy up to logarithmic factors for four-cycle-free matrices and recovers Szemerédi–Trotter-type discrepancy in geometric set systems. Collectively, these results advance the understanding of how bounded γ2-norms constrain combinatorial density and have concrete implications for Zarankiewicz problems and incidence geometry in semilinear and geometric contexts.

Abstract

The $γ_2$-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 $γ_2(M)<γ$ and $M$ contains no $t\times t$ all-ones submatrix, then $M$ contains $O_{γ,t}(m+n)$ one entries. In other words, graphs of bounded $γ_2$-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.

Factorization norms and Zarankiewicz problems

TL;DR

This work studies the γ2-norm (max-norm) of Boolean matrices, linking spectral decompositions to combinatorial structure. It establishes that four-cycle-free Boolean matrices with degeneracy d satisfy γ2(M) = Θ(√d), enabling degree-boundedness results: matrices with bounded γ2-norm and no large all-ones submatrices have few ones, and various decompositions (into thin blocky matrices) are equivalent to bounded degeneracy. These insights yield Zarankiewicz-type incidence bounds for point–polytope configurations, showing that K_{t,t}-free incidence graphs between n points and polytopes from POL(H) have edges bounded by O_{s,t}(n (log n)^{O(d)}) in semilinear settings, with the log-exponent depending only on dimension. The paper also connects to discrepancy theory via the Matoušek–Nikolov–Talwar inequality, demonstrating that γ2 controls hereditary discrepancy up to logarithmic factors for four-cycle-free matrices and recovers Szemerédi–Trotter-type discrepancy in geometric set systems. Collectively, these results advance the understanding of how bounded γ2-norms constrain combinatorial density and have concrete implications for Zarankiewicz problems and incidence geometry in semilinear and geometric contexts.

Abstract

The -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 is an Boolean matrix such that and contains no all-ones submatrix, then contains one entries. In other words, graphs of bounded -norm are degree bounded. This addresses a conjecture of Hambardzumyan, Hatami, and Hatami for locally sparse matrices. We prove that if is a -free incidence graph of points and homothets of a polytope in , then the average degree of is . This is sharp up the notations. In particular, we prove a more general result on semilinear graphs, which greatly strengthens the work of Basit, Chernikov, Starchenko, Tao, and Tran.

Paper Structure

This paper contains 14 sections, 21 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Matrices of bounded max-norm
  3. Communication complexity
  4. Zarankiewicz problem
  5. Discrepancy theory
  6. Preliminaries
  7. Combinatorics of matrices
  8. Linear algebra notation
  9. The max-norm
  10. Blocky matrices
  11. Basic results
  12. Sparse matrices
  13. Point-polytope incidences
  14. Concluding remarks

Key Result

Theorem 1.2

Let $t\geq 2$ be an integer and $\gamma>0$. Then there exists $d=d(\gamma,t)$ such that every $m\times n$ Boolean matrix $M$ with no $t\times t$ all-ones submatrix and $\gamma_2(M)\leq \gamma$ contains at most $d(m+n)$ one entries.

Figures (2)

  • Figure 1: A blocky matrix, where blank entries denote zeros. Any row and column permutation is also a blocky matrix.
  • Figure 2: A $K_{2,2}$-free configuration of points and elements of $\mathop{\mathrm{POL}}\nolimits(\mathcal{H})$, where $\mathcal{H}$ contains 6 half-planes.

Theorems & Definitions (48)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • proof
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 1: Degeneracy
  • ...and 38 more