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 $n\geqslant 4k\binom{k}{s}r$.