Factorization norms and an inverse theorem for MaxCut

We prove that Boolean matrices with bounded <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>γ</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>γ</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:math> -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 <i>G</i> with <i>m</i> edges has a cut of size at least <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mfrac><mml:mi>m</mml:mi> <mml:mn>2</mml:mn></mml:mfrac> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mrow> <mml:msqrt><mml:mrow><mml:mn>8</mml:mn> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:msqrt> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn></mml:mrow> <mml:mn>8</mml:mn></mml:mfrac> </mml:mrow> </mml:math> , with equality achieved by complete graphs with an odd number of vertices. To contrast this, we prove that if the MaxCut of <i>G</i> is at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mfrac><mml:mi>m</mml:mi> <mml:mn>2</mml:mn></mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>O</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:msqrt><mml:mi>m</mml:mi></mml:msqrt> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> , then <i>G</i> must contain a clique of size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:msqrt><mml:mi>m</mml:mi></mml:msqrt> <mml:mo>)</mml:mo></mml:mrow> </mml:math> .