Induced Tur\'an problem in bipartite graphs

The classical extremal function for a graph $H$, $ex(K_n, H)$ is the largest number of edges in a subgraph of $K_n$ that contains no subgraph isomorphic to $H$. Note that defining $ex(K_n, H-ind)$ by forbidding induced subgraphs isomorphic to $H$ is not very meaningful for a non-complete $H$ since one can avoid it by considering a clique. For graphs $F$ and $H$, let $ex(K_n, \{F, H-ind\})$ be the largest number of edges in an $n$-vertex graph that contains no subgraph isomorphic to $F$ and no induced subgraph isomorphic to $H$. Determining this function asymptotically reduces to finding either $ex(K_n, F)$ or $ex(K_n, H)$, unless $H$ is a biclique or both $F$ and $H$ are bipartite. Here, we consider the bipartite setting, $ex(K_{n,n}, \{F, H-ind\})$ when $K_n$ is replaced with $K_{n,n}$, $F$ is a biclique, and $H$ is a bipartite graph. Our main result, a strengthening of a result by Sudakov and Tomon, implies that for any $d\geq 2$ and any $K_{d,d}$-free bipartite graph $H$ with each vertex in one part of degree either at most $d$ or a full degree, so that there are at most $d-2$ full degree vertices in that part, one has $ex(K_{n,n}, \{K_{t,t}, H-ind\}) = o(n^{2-1/d})$. This provides an upper bound on the induced Tur\'an number for a wide class of bipartite graphs and implies in particular an extremal result for bipartite graphs of bounded VC-dimension by Janzer and Pohoata.