The case $s=1$ is discussed in
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 $k$-subsets is considered in
Peter Frankl and Andrey Kupavskii. (2016). Families with no $s$ pairwise disjoint sets. Journal of the London Mathematical Society. 95. arxiv:1607.06122
Peter Frankl (2018). On families of sets without $k$ pairwise disjoint members, Journal of Combinatorial Theory, Series B, Volume 130, Pages 92-97 doi:10.1016/j.jctb.2017.10.002