A Proof of a Frankl–Kupavskii Conjecture on Intersecting Families

A family \(\mathcal {F} \subset \mathcal {P}(n)\) is r-wise k-intersecting if \(|A_1 \cap \dots \cap A_r| \ge k\) for any \(A_1, \dots , A_r \in \mathcal {F}\). It is easily seen that if \(\mathcal {F}\) is r-wise k-intersecting for \(r \ge 2\), \(k \ge 1\) then \(|\mathcal {F}| \le 2^{n-1}\). The problem of determining the maximum size of a family \(\mathcal {F}\) that is both \(r_1\)-wise \(k_1\)-intersecting and \(r_2\)-wise \(k_2\)-intersecting was raised in 2019 by Frankl and Kupavskii (Combinatorica 39:1255–1266, 2019). They proved the surprising result that, for \((r_1,k_1) = (3,1)\) and \((r_2,k_2) = (2,32)\) then this maximum is at most \(2^{n-2}\), and conjectured the same holds if \(k_2\) is replaced by 3. In this paper we shall not only prove this conjecture but we shall also determine the exact maximum for \((r_1,k_1) = (3,1)\) and \((r_2,k_2) = (2,3)\) for all n.