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?
The case $s=1$ is Erdős hypergraph mathcing conjecture from
Paul Erdős (1965). A problem on independent $r$-tuples. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 8 (1965), 93–95. users.renyi.hu/~p_erdos/1965-01.pdf
A recent paper about it is
Peter Frankl (2017) Proof of the Erdős matching conjecture in a new range, Israel Journal of Mathematics 222(1), pp 421-430 doi:10.1007/s11856-017-1595-7
The generalization you are asking about is studied in
Christos Pelekis, Israel Rocha (2017), A generalization of Erdős' matching conjecture. arxiv:1710.04633
In the notation of this paper you are asking for the maximum number of edges in a $k$-uniform hypergraph, such that its $s$-matching number is strictly less than $r$. Their abstract says that they identify a collection of candidate solutions, and show that it contains the optimum when $n\geqslant 4k\binom{k}{s}r$.
