The case $s=1$ is discussed in
Frankl, Peter & Kupavskii, Andrey. (2016). Families with no $s$ pairwise disjoint sets. Journal of the London Mathematical Society. 95. arxiv:1607.06122
The case $s=1$ is discussed in
Frankl, Peter & Kupavskii, Andrey. (2016). Families with no $s$ pairwise disjoint sets. Journal of the London Mathematical Society. 95. arxiv:1607.06122