Cross-intersecting families with covering number constraints
Yandong Bai, Haoyun Gu
TL;DR
The paper addresses maximizing the sum $|\mathcal{F}|+|\mathcal{G}|$ for cross-intersecting uniform families under general covering-number constraints, placing the problem within a unified stability framework. It develops shifting techniques to reduce to initial subfamilies and introduces extremal constructions $\mathcal{M}^t(n,a,b)$, $\mathcal{M}_s^t(n,a,b)$, $\mathcal{H}^t(n,a,b)$, and $\mathcal{H}_s^t(n,a,b)$ to capture the maxima under four regimes (1)–(4). The main results establish sharp upper bounds and exact extremal structures for all cases with $a \ge b+t-1$ and $n \ge \max\{a+b, bt\}$, providing complete characterizations up to isomorphism. These findings unify classical theorems (EKR, Hilton–Milner, Frankl–Tokushige) with recent advances (Frankl 2024, Frankl–Wang 2025), offering a comprehensive stability framework for cross-intersecting uniform families under covering-number constraints.
Abstract
Two families $\mathcal{F}$ and $\mathcal{G}$ are cross-intersecting if every set in $\mathcal{F}$ intersects every set in $\mathcal{G}$. The covering number $τ(\mathcal{F})$ of a family $\mathcal{F}$ is the minimum size of a set that intersects every member of $\mathcal{F}$. In 1992, Frankl and Tokushige determined the maximum of $|\mathcal{F}| + |\mathcal{G}|$ for cross-intersecting families $\mathcal{F} \subset \binom{[n]}{a}$ and $\mathcal{G} \subset \binom{[n]}{b}$ that are non-empty (covering number at least 1) and also characterized the extremal configurations. This seminar result was recently extended by Frankl (2024) and Frankl and Wang (2025) to cases where both families are non-trivial (covering number at least 2), and where one is non-empty and the other non-trivial, respectively. In this paper, we establish a unified stability hierarchy for cross-intersecting families under general covering number constraints. We determine the maximum of $|\mathcal{F}| + |\mathcal{G}|$ for cross-intersecting families $\mathcal{F} \subset \binom{[n]}{a}$ and $\mathcal{G} \subset \binom{[n]}{b}$ with the following covering number constraints: (1) $τ(\mathcal{F}) \geq s$ and $τ(\mathcal{G}) \geq t$; (2) $τ(\mathcal{F}) = s$ and $τ(\mathcal{G}) \geq t \geq 2$; (3) $τ(\mathcal{F}) \geq s$ and $τ(\mathcal{G}) = t$; (4) $τ(\mathcal{F}) = s$ and $τ(\mathcal{G}) = t$; provided $a \geq b + t - 1$ and $n \geq \max\{a + b, bt\}$. The corresponding extremal families achieving the upper bounds are also characterized.