Factorization norms and an inverse theorem for MaxCut

TL;DR

This paper proves structural properties of Boolean matrices with bounded norms, verifies a conjecture, and establishes an inverse theorem for MaxCut, linking spectral properties to graph structure and MaxCut bounds.

Contribution

It introduces new structural results for Boolean matrices with bounded $oldsymbol{ extgamma_2}$-norm and proves an inverse theorem for MaxCut relating MaxCut size to clique size.

Findings

Boolean matrices with bounded norms contain large uniform submatrices

Graphs with small MaxCut contain large cliques

Verification of a conjecture on matrix structure

Abstract

We prove that Boolean matrices with bounded $\gamma_2$-norm or bounded normalized trace norm must contain a linear-sized all-ones or all-zeros submatrix, verifying a conjecture of Hambardzumyan, Hatami, and Hatami. We also present further structural results about Boolean matrices of bounded $\gamma_2$-norm and discuss applications in communication complexity, operator theory, spectral graph theory, and extremal combinatorics. As a key application, we establish an inverse theorem for MaxCut. A celebrated result of Edwards states that every graph $G$ with $m$ edges has a cut of size at least $\frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}$, with equality achieved by complete graphs with an odd number of vertices. To contrast this, we prove that if the MaxCut of $G$ is at most $\frac{m}{2}+O(\sqrt{m})$, then $G$ must contain a clique of size $\Omega(\sqrt{m})$.