Subgraphs of a bipartite graph $H$ as the intersection of two copies of $H$ in $K_{n,n}$. Let $H$ be a bipartite graph. Is it necessarily the case that every subgraph of $H$ can appear as the intersection of two copies of $H$ in $K_{n,n}$ for large enough $n$? (Here $K_{n,n}$ is the complete bipartite graph with $n$ vertices in each class.) Give a proof or a counterexample.
No idea how to approach this at all and even on what the answer might be. Any help appreciated!

The answer is negative for $H=C_4$ and its subgraph $G$ which equals $H$ without an edge. Indeed, if a copy of $G$ is an intersection of two copies of $H$ then both copies share the same four vertices and it is easy to check that the intersection of the copies of $H$ cannot be a copy of $G$.  
On the other hand, the answer is positive for each induced subgraph $G$ of $H$. Indeed, let $H’$ be a graph $H$ in which we cloned each vertex which in not in $G$. So $H’$ is a union of two copies of $H$ whose intersection is a copy of $G$. Since $H’$ is bipartite, it is a subgraph of $K_{p,p}$ for $p\le 2n_H-n_G$, where $n_H$ and $n_G$ are the numbers of vertices of graphs $H$ and $G$, respectively.
