ALIBABA 2021 Problem 3

Given positive integers $k \ge 2$ and $m$ sufficiently large. Let $\mathcal{F}_m$ be the infinite family of all the (not necessarily square) binary matrices which contain exactly $m$ 1's. Denote by $f(m)$ the maximum integer $L$ such that for every matrix $A \in \mathcal{F}_m$ , there always exists a binary matrix $B$ of the same dimension such that (1) $B$ has at least $L$ 1-entries; (2) every entry of $B$ is less or equal to the corresponding entry of $A$ ; (3) $B$ does not contain any $k \times k$ all-1 submatrix. Show the equality $$\lim_{m \to \infty} \frac{\ln f(m)}{\ln m}=\frac{k}{k+1}.$$