Let \( n \geq t \geq 1 \) and \( m \geq 2 \) be integers. A collection of \( m \) families \( \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]} \) is said to be pairwise cross t-intersecting if for any \( A_i \in \mathcal{A}_i \) and \( A_j \in \mathcal{A}_j \) with \( i \neq j \), we have \( |A_i \cap A_j| \geq t \).In this talk, we present a sharp upper bound on the sum of their sizes:\[\sum_{i=1}^{m}|\mathcal{A}_i|\leq \max\left\{ \sum_{k=t}^{n}\binom{n}{k} + m - 1,\  m \cdot M(n,t) \right\},\]where \( M(n,t) \) denotes the maximum size of a \( t \)-intersecting family in \( 2^{[n]} \). Furthermore, for \( t > 1 \), we provide a complete characterization of the extremal families that achieve this bound.

Our result generalizes the classical theorem of Katona (1964) for a single family and extends a theorem of Frankl and Wong (2021) for two families. It can also be viewed as a non-uniform version of a recent theorem of Li and Zhang (2025).

吴勇江，中南大学博士，导师是冯立华教授。主要研究方向是代数图论和极值集合。目前，在《J. Combin. Theory, Ser. A》、《Adv. in Appl. Math.》、《Discrete Math.》、《Linear Algebra Appl.》等知名期刊上发表论文多篇。