Biholes in balanced bipartite graphs

A bihole in a bipartite graph $G$ with partite sets $A$ and $B$ is an independent set $I$ in $G$ with $\mathrm{\mid }I\cap A\mathrm{\mid }=\mathrm{\mid }I\cap B\mathrm{\mid }$. We prove lower bounds on the largest order of biholes in balanced bipartite graphs subject to conditions involving the vertex degrees and the average degree.