# Upper bound on number of bipartite subgraphs

Given a graph with $n$ vertices and $m$ edges, and fix positive integers $a+b\leq n$. What are some upper bounds on the number of (ordered) pairs of subsets of vertices $(A,B)$ such that $|A|=a,|B|=b$, and any vertex in $A$ has an edge to any vertex in $B$?

If we only use the variables $n,a,b$, then it is possible that the whole graph is complete, and so the number of such pairs is $\binom{n}{a}\cdot\binom{n-a}{b}=\frac{n!}{a!b!(n-a-b)!}$. But can we have a bound that takes into account the number of edges $m$?

All right, you already have some finite upper bound :-) I'll give exact bounds for some special cases. Let $k = k(n, m, a, b)$ be maximum number of desired pairs of subsets.
If $m = \binom{n}{2}$ then as far as you know $k = \frac{n!}{a!b!(n - a - b)!}$. If $m < ab$, then $k = 0$. If $m = ab$, then $k = 1 + [a = b]$. If $a = b = 1$, then $k = 2m$.
If $a = 1$ and $b > 1$, then each vertex of degree $d$ gives $\binom{d}{b}$ subgraphs. It is easy to see that universal vertex is the best choice in sense of relation between edges used and subgraphs gain. Graph can have $\ell$ universal vertices if $\binom{\ell}{2} + \ell (n - \ell) \le m$, i. e., if $\ell \le n - \frac12 - \sqrt{n^2 - n + \frac14 - 2m}$ for $m < \binom{n}{2}$. After getting maximum number of universal vertices it is better to give all remaining $r = m - \binom{\ell}{2} - \ell (n - \ell)$ edges to one of remaining vertices. Then $k = \ell\binom{n}{b} + \binom{\ell + r}{b} + (n - 1 - \ell)\binom{\ell}{b}$ for $\ell = \left\lfloor n - \frac12 - \sqrt{n^2 - n + \frac14 - 2m}\right\rfloor$ and $r = m - \binom{\ell}{2} - \ell (n - \ell)$.