Let $[n]:=\{1,\dots,n\}$ and $0\leq p_n\leq n$. Fix any subset $A_n$ of $[n]$ with $p_n$ elements. The number of subsets $B$ of $[n]$ with $p_n$ elements that are disjoint from $A$ is $\binom{n-p_n}{p_n}$ and so the number of subsets $B$ with $p_n$ elements that have a non-empty intersection with $A$ is given by $\binom{n}{p_n}-\binom{n-p_n}{p_n}$ which asymptotically behaves like $\binom{n}{p_n}$ if $\sqrt{n}\ll p_n$. Precisely: $$\lim_{n\to\infty}\frac{\binom{n}{p_n}-\binom{n-p_n}{p_n}}{\binom{n}{p_n}}=\begin{cases}1 &\text{if }\sqrt{n}\ll p_n\\ 0&\text{if }p_n\ll\sqrt{n}\\ e^{-a^2}&\text{if }\lim_{n\to\infty}\frac{p_n}{\sqrt{n}}=a\end{cases}$$ To put it differently, any two subsets of $[n]$ with $p_n$ elements typically have non-empty intersection if $p_n$ grows faster than $\sqrt{n}$.

Now, I am interested in the typical number of elements in their intersection if $p_n$ grows faster than $\sqrt{n}$. My feeling would be that this number also grows with $n$. Precisely, I believe (numerics support this) that if $q_n$ grows slower than $p_n/\sqrt{n}$ then eventually all sets $A_n$, $B_n$ with $p_n$ elements each share at least $q_n$ elements. To put it in formulas, if $A_n$ is any set with $p_n$ elements, then $$\lim_{n\to\infty }\frac{\lvert\{B_n\subset[n]~~\lvert~~\lvert B_n\lvert=p_n\text{ and }\lvert A_n\cap B_n\lvert\geq q_n\}\rvert}{\binom{n}{p_n}}=1.$$

Since there is no exact combinatorial count (only in terms of generalised hypergeometric functions) on the number of $B_n$ of a given size that share at least $q_n$ elements with some fixed sets, I would need some good lower bound on it. Has anybody seen those asymptotics worked out? Any references would be welcome.