I just wanted to know, is there any result know about the following generalization of Erdős-Ko-Rado theorem?

Let $n, k, r, s$ be positive integers. We call a family $\mathcal{F}$ of k-subsets of the set $\{1,\ldots, n\}$, $(r, s)$-intersection family if among every $r$ elements of $\mathcal{F}$ at least two of them has $s$- elements in their intersection. What is the maximum size of such a family?