Large monochromatic submatrices in bounded max-norm matrices

Prove that for every constant C greater than 0 there exists c greater than 0 such that every m by n binary matrix M with gamma_2(M) at most C contains either an all-ones or an all-zeros submatrix of dimensions at least cm by cn.

Background

The paper notes that the blocky-decomposition conjecture would immediately imply a linear-sized constant submatrix in every bounded-gamma_2 binary matrix. It proposes whether this consequence might be established directly, without proving the stronger decomposition conjecture.

The proposed statement asks for a genuinely dense rectangular monochromatic structure whose dimensions are proportional to those of the entire matrix. It remains unresolved in the paper.

References

For every $C>0$ there exists $c>0$ such that the following holds. Let $M$ be an $m\times n$ binary matrix such that $\gamma_2(M)\leq C$. Then $M$ contains a $cm\times cn$ sized all-ones or all-zeros submatrix.

— Factorization norms and Zarankiewicz problems  (2502.18429 - Tomon, 25 Feb 2025) in Section "Open problems"