The case $s=1$ is discussed inErdős 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 same problem without the restriction to $k$-subsetsgeneralization you are asking about is consideredstudied in
Peter Frankl and Andrey Kupavskii.Christos Pelekis, Israel Rocha (20162017). Families with no $s$ pairwise disjoint sets. Journal, A generalization of the London Mathematical Society. 95Erdős' matching conjecture. arxiv:1607.06122arxiv:1710.04633
Peter Frankl (2018). On familiesIn the notation of sets withoutthis paper you are asking for the maximum number of edges in a $k$ pairwise disjoint members-uniform hypergraph, Journalsuch that its $s$-matching number is strictly less than $r$. Their abstract says that they identify a collection of Combinatorial Theory, Series B, Volume 130candidate solutions, Pages 92-97and show that it contains the optimum when doi:10.1016/j.jctb.2017.10.002$n\geqslant 4k\binom{k}{s}r$.