Is there any result known about the following generalization of the Erdős-Ko-Rado theorem?

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