Multi-part cross-intersecting families

TL;DR

This paper investigates the structure and maximum sizes of multi-part cross-intersecting families of sets, generalizing classical single-part results through graph-theoretic characterizations.

Contribution

It introduces a characterization of maximum multi-part cross-intersecting families using vertex-transitive graphs, extending known single-part theorems.

Findings

Determined the maximum sizes of multi-part cross-intersecting families.

Characterized the structure of these families in the multi-part setting.

Generalized classical results to a broader multi-part context.

Abstract

Let $\mathcal{A}\subseteq{[n]\choose a}$ and $\mathcal{B}\subseteq{[n]\choose b}$ be two families of subsets of $[n]$, we say $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\neq \emptyset$ for all $A\in\mathcal{A}$, $B\in\mathcal{B}$. In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton's and Frankl--Tohushige's on cross-intersecting families in the single-part setting.