A Product Version of the Hilton-Milner-Frankl Theorem

TL;DR

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

Contribution

It establishes the maximum product of sizes for two non-trivial cross t-intersecting families, generalizing the Hilton-Milner-Frankl Theorem to a product setting.

Findings

Maximum product achieved by specific family structures

Conditions on n and k for the theorem to hold

Extension of classical intersecting family results

Abstract

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross $t$-intersecting if $|F\cap G|\geq t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $|\cap \{F\colon F\in \mathcal{F}\}|<t$, $|\cap \{G\colon G\in\mathcal{G}\}|<t$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross $t$-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq 4(t+2)^2k^2$, $k\geq 5$, which is a product version of the Hilton-Milner-Frankl Theorem.