# Factorization norms and an inverse theorem for MaxCut

## TL;DR

The paper proves a conjecture about Boolean matrices and applies it to show that graphs with small MaxCut must contain large cliques.

## Contribution

The paper verifies a conjecture about Boolean matrices and establishes a new inverse theorem for MaxCut.

## Key findings

- Boolean matrices with bounded γ₂-norm contain large all-ones or all-zeros submatrices.
- Graphs with small MaxCut must contain large cliques, proving an inverse theorem for MaxCut.

## Abstract

We prove that Boolean matrices with bounded \documentclass[12pt]{minimal}
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\begin{document}$$\gamma _2$$\end{document}γ2-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 \documentclass[12pt]{minimal}
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\begin{document}$$\gamma _2$$\end{document}γ2-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 \documentclass[12pt]{minimal}
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\begin{document}$$\frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}$$\end{document}m2+8m+1-18, 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 \documentclass[12pt]{minimal}
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\begin{document}$$\frac{m}{2}+O(\sqrt{m})$$\end{document}m2+O(m), then G must contain a clique of size \documentclass[12pt]{minimal}
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\begin{document}$$\Omega (\sqrt{m})$$\end{document}Ω(m).