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.

如果\( | A_1 \cap \ dots \ cap A_r| \ge k \ )对于任何\ ( A_1, \dots , A_r \in \mathcal {F}\)。很容易看出，如果\(\mathcal {F}\)与\(r \ge 2\)、\(k \ge 1\)呈r方向k相交，则\(|\mathcal {F}| \le 2^{n-1}\)。确定族\(\mathcal {F}\)的最大大小的问题，该族 \(\mathcal {F}\) 既是\(r_1\)方向\(k_1\)相交，又是\(r_2\)方向\(k_2\)相交-相交由 Frankl 和 Kupavskii 于 2019 年提出（Combinatorica 39:1255–1266, 2019）。他们证明了令人惊讶的结果，对于\((r_1,k_1) = (3,1)\)和\((r_2,k_2) = (2,32)\)那么这个最大值最多为\(2^{ n-2}\) ，并且推测如果\(k_2\)被 3 代替，同样成立。在本文中，我们不仅要证明这个猜想，而且还要确定\((r_1,k_1) =的确切最大值(3,1)\)和\((r_2,k_2) = (2,3)\)对于所有n。