$r$-cross $t$-intersecting families via necessary intersection points

TL;DR

This paper generalizes the Hilton-Milner theorem to $r$-cross $t$-intersecting families, determining maximum sizes and measures for uniform and non-uniform families, extending previous partial results.

Contribution

It provides a comprehensive generalization of the Hilton-Milner theorem for $r$-cross $t$-intersecting families, including uniform and arbitrary families, with measure-based results.

Findings

Determined maximum sum of sizes for $r$-cross $t$-intersecting families.

Extended results to families with mixed uniformities.

Unified measure-based framework for these families.

Abstract

Given integers $r\geq 2$ and $n,t\geq 1$ we call families $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq\mathscr{P}([n])$ $r$-cross $t$-intersecting if for all $F_i\in\mathcal{F}_i$, $i\in[r]$, we have $\vert\bigcap_{i\in[r]}F_i\vert\geq t$. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. In particular, we determine the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for $r$-cross $t$-intersecting families in the cases when these are $k$-uniform families or arbitrary subfamilies of $\mathscr{P}([n])$. Only some special cases of these results had been proved before. We obtain the aforementioned theorems as instances of a more general result that considers measures of $r$-cross $t$-intersecting families. This also provides the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for families of possibly mixed uniformities $k_1,\ldots,k_r$.