Bipartite independence number in graphs with bounded maximum degree

We consider a natural, yet seemingly not much studied, extremal problem in bipartite graphs. A bi-hole of size $t$ in a bipartite graph $G$ is a copy of $K_{t, t}$ in the bipartite complement of $G$. Let $f(n, \Delta)$ be the largest $k$ for which every $n \times n$ bipartite graph with maximum degree $\Delta$ in one of the parts has a bi-hole of size $k$. Determining $f(n, \Delta)$ is thus the bipartite analogue of finding the largest independent set in graphs with a given number of vertices and bounded maximum degree. Our main result determines the asymptotic behavior of $f(n, \Delta)$. More precisely, we show that for large but fixed $\Delta$ and $n$ sufficiently large, $f(n, \Delta) = \Theta(\frac{\log \Delta}{\Delta} n)$. We further address more specific regimes of $\Delta$, especially when $\Delta$ is a small fixed constant. In particular, we determine $f(n, 2)$ exactly and obtain bounds for $f(n, 3)$, though determining the precise value of $f(n, 3)$ is still open.