A result for hemi-bundled cross-intersecting families
Yongjiang Wu, Lihua Feng, Yongtao Li
TL;DR
The paper advances the theory of cross-intersecting families by generalizing Frankl's hemi-bundled bound to the regime where the $t$-intersecting family $\mathcal{F}$ has size at least $r$, yielding precise upper bounds for $|\mathcal{F}|+|\mathcal{G}|$ in two regimes and explicit extremal structures. It also sharpens Katona's classical stability for $s$-union families in the non-uniform setting, introducing Katona-type extremal families and proving stability results that recover and extend prior work of Frankl, Kupavskii, and Wu. The main method combines shifting with careful case analysis and lexicographic extremal techniques to derive both bounds and complete descriptions of extremal families in the cross-intersecting and $s$-union contexts. These results yield streamlined proofs and new extremal configurations for several known theorems, including sharp versions of Frankl–Wang and related diversity results, and provide a unified framework for stability phenomena in extremal set theory. Overall, the work deepens understanding of how intersection, cross-intersection, and union-constraints interact in both uniform and non-uniform families, with potential impact on related combinatorial optimization and probabilistic methods.
Abstract
Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, 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 $k \geq 1, t\ge 0$ and $n \geq 2 k+t$, if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, in which $\mathcal{F}$ is non-empty and $(t+1)$-intersecting, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. This bound can be attained when $\mathcal{F}$ consists of a single set. In this paper, we generalize this result under the constraint $|\mathcal{F}| \geq r$ for every $r\leq n-k-t+1$. 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.