Finding Big Blocks in Small‑Norm Boolean Matrices

The study shows that if a matrix filled with 0s and 1s has either a small γ₂‑norm or a small normalized trace norm, it must hide a large square of all 1s or all 0s. This confirms a claim made by Hambardzumyan, Hatami, and Hatami. The researchers also explore other patterns that arise when Boolean matrices keep their γ₂‑norm low, and they link these findings to fields such as communication limits, operator theory, graph spectra, and combinatorial extremes.A notable consequence is an “inverse” version of a classic graph cut result by Edwards. Edwards proved that any graph with m edges contains a cut of size at least (m/2) plus roughly √(8m+1)/8, and this bound is tight for complete graphs with an odd number of vertices. The new work flips the idea: if a graph’s maximum cut is only slightly larger than m/2—specifically, no more than (m/2)+O(√m)—then the graph must contain a clique whose size grows like Ω(√m). In other words, having a very small maximum cut forces the graph to hold a large complete subgraph.