A result for hemi-bundled cross-intersecting families

Abstract

Two families $F$ and $G$ are called cross-intersecting if for every $F\in F$ and $G\in G$, the intersection $F\cap G$ is non-empty. It is significant to determine the maximum sum of sizes of cross-intersecting families under the additional assumption that one of the two families is intersecting. Such a pair of families is said to be hemi-bundled. In particular, Frankl (2016) proved that for $k\ge 1,t\ge 0$ and $n\ge 2k+t$, if $F\subseteq \left(\begin{array}{c}\left[n\right]\\ k+t\end{array}\right)$ and $G\subseteq \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$ are cross-intersecting families, in which $F$ is non-empty and $(t+1)$-intersecting, then $\left|F\right|+\left|G\right|\le \left(\begin{array}{c}n\\ k\end{array}\right)-\left(\begin{array}{c}n-k-t\\ k\end{array}\right)+1$. This bound is achieved when $F$ consists of a single set. In this paper, we generalize this result under the constraint $\left|F\right|\ge r$ for every $r\le n-k-t+1$. Moreover, we investigate the stability results of Katona's theorem for non-uniform families with the s-union property. Our result extends the stabilities established by Frankl (2017) and Li and Wu (2024). As applications, we revisit a recent result of Frankl and Wang (2024) as well as a result of Kupavskii (2018). Furthermore, we determine the extremal families in these two results.