Large empty rectangles from bounded induced four-cycle-free density

Prove that for every positive k there exist positive epsilon and c such that every bipartite graph with vertex classes A and B of size n, maximum degree at most epsilon n, and no induced four-cycle-free subgraph of average degree greater than k contains subsets X of A and Y of B with equal size at least cn and with no edges between X and Y.

Background

The paper states that the preceding monochromatic-submatrix conjecture would follow from this combinatorial graph problem, together with the paper's theorem characterizing the gamma_2-norm of four-cycle-free Boolean matrices in terms of degeneracy.

The condition excludes dense induced four-cycle-free subgraphs while imposing a linear maximum-degree restriction. The desired conclusion is a linearly sized empty bipartite rectangle, which corresponds to an all-zeros submatrix in the associated adjacency matrix.

References

For every $k>0$ there exist $\varepsilon,c>0$ such that the following holds. Let $G$ be a bipartite graph with vertex classes $A,B$ of size $n$. Assume that the maximum degree of $G$ is at most $\varepsilon n$, and that $G$ contains no induced four cycle-free subgraph of average degree more than $k$. Then there exists $X\subset A$ and $Y\subset B$ such that $|X|=|Y|\geq cn$ and there are no edges between $X$ and $Y$.

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