Finding Big Blocks in Small‑Norm Boolean Matrices

The latest study demonstrates a striking property of binary matrices.
If a matrix filled with 0s and 1s has either:

• a small γ₂‑norm, or
• a small normalized trace norm,

then it must contain a large square submatrix that is either all 1s or all 0s.

This finding confirms a claim originally made by Hambardzumyan, Hatami, and Hatami.
Beyond the square pattern, the researchers investigate additional structures that emerge when Boolean matrices maintain a low γ₂‑norm. They connect these patterns to several areas:

• Communication limits
• Operator theory
• Graph spectra
• Combinatorial extremes

An Inverse Graph Cut Result

A notable implication of the work is an “inverse” version of a classic graph cut theorem by Edwards:

• Edwards’ original result:
  For any graph with m edges, there exists a cut of size at least
  [ \frac{m}{2} + \sqrt{\frac{8m+1}{8}} ] This bound is tight for complete graphs with an odd number of vertices.

• The new inverse theorem:
  If a graph’s maximum cut is only slightly larger than m/2—specifically, no more than
  [ \frac{m}{2} + O(\sqrt{m}) ] —then the graph must contain a clique whose size grows on the order of
  [ \Omega(\sqrt{m}) ]

In plain terms, a graph with an exceptionally small maximum cut is forced to host a large complete subgraph.

Takeaway

The study not only solidifies theoretical bounds on matrix norms but also bridges them to tangible graph properties, offering new tools for analyzing combinatorial structures and communication systems.