A family F ⊂ P(n) is r-wise k-intersecting if |A1 ∩ · · · ∩ Ar| ≥ k for any A1, . . . , Ar ∈ F. It is easily seen that if F is r-wise k-intersecting for r ≥ 2, k ≥ 1 then |F| ≤ 2^(n−1) . The problem of determining the maximal size of a family F that is both r1-wise k1-intersecting and r2-wise k2-intersecting was raised in 2019 by Frankl and Kupavskii. They proved the surprising result that, for (r1, k1) = (3, 1) and (r2, k2) = (2, 32) then this maximum is at most 2^(n−2) , and conjectured the same holds if k2 is replaced by 3. In this talk I shall not only prove this conjecture but also determine the exact maximum for (r1, k1) = (3, 1) and (r2, k2) = (2, 3) for all n.
