On the maximum size of 3-uniform U(s, q) families

Let n, k, s, q be integers, n > q > k, sk > q and s > 2. A family F ⊂ is U(s, q) if |F1∪···∪Fs| 6 q for any F1, . . ., Fs ∈ F. This notion was introduced recently by Frankl and Kupavskii (2021) and it generalizes the property of a family being t-intersecting, or having no more than s pairwise disjoint members. Frankl and Kupavskii (2021) formulated a problem of finding the maximum size of U(s, q) families. In this paper, we study the k = 3 case extensively, and determine the maximum size of U(s, 2s+t) families for all s > s0(t) and all meaningful n and q. In particular, we prove a conjecture proposed by Frankl and Kupavskii (2021) on the maximum size of 3-uniform U(s, q) families.