A Product Version of the Hilton-Milner Theorem

TL;DR

This paper extends the Hilton-Milner Theorem by determining the maximum product of sizes for two non-trivial cross-intersecting families of k-subsets, providing a new product version of the classical combinatorial result.

Contribution

It introduces a novel product version of the Hilton-Milner Theorem, establishing the maximum product for non-trivial cross-intersecting families of k-subsets.

Findings

Maximum product of sizes determined for n ≥ 4k, k ≥ 8

Characterization of extremal families provided

Extends classical Hilton-Milner Theorem to a product setting

Abstract

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross-intersecting if $F\cap G\neq \emptyset$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $\cap \{F\colon F\in \mathcal{F}\}=\emptyset=\cap \{G\colon G\in \mathcal{G}\}$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq 4k$, $k\geq 8$, which is a product version of the classical Hilton-Milner Theorem.