Cross-intersecting families with covering number constraints

TL;DR

This paper unifies and extends stability results for cross-intersecting families with various covering number constraints, determining maximum sizes and extremal configurations under these conditions.

Contribution

It establishes a comprehensive stability hierarchy for cross-intersecting families with general covering number constraints, generalizing previous results.

Findings

Maximum sum of sizes for cross-intersecting families under covering constraints.

Characterization of extremal families achieving maximum sums.

Extension of known results to broader covering number scenarios.

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 $\tau(\mathcal{F})$ 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) $\tau(\mathcal{F}) \geq s$ and $\tau(\mathcal{G}) \geq t$; (2) $\tau(\mathcal{F}) = s$ and $\tau(\mathcal{G}) \geq t \geq 2$; (3) $\tau(\mathcal{F}) \geq s$ and $\tau(\mathcal{G}) = t$; (4) $\tau(\mathcal{F}) = s$ and $\tau(\mathcal{G}) = t$; provided $a \geq b + t - 1$ and $n \geq \max\{a + b, bt\}$. The corresponding extremal families achieving the upper bounds are also characterized.