# Non-trivially intersecting multi-part families

## TL;DR

This paper investigates the maximum size of non-trivially intersecting set families in a multi-part setting, extending classical combinatorial results and disproving a recent conjecture.

## Contribution

It determines the maximum size of such families when part sizes are large, advancing understanding in multi-part intersection problems.

## Key findings

- Maximum size of non-trivially intersecting families determined for large parts
- Disproves a conjecture of Alon and Katona
- Extends classical intersection theorems to multi-part settings

## Abstract

We say a family of sets is intersecting if any two of its sets intersect, and we say it is trivially intersecting if there is an element which appears in every set of the family. In this paper we study the maximum size of a non-trivially intersecting family in a natural "multi-part" setting. Here the ground set is divided into parts, and one considers families of sets whose intersection with each part is of a prescribed size. Our work is motivated by classical results in the single-part setting due to Erd\H{o}s, Ko and Rado, and Hilton and Milner, and by a theorem of Frankl concerning intersecting families in this multi-part setting. In the case where the part sizes are sufficiently large we determine the maximum size of a non-trivially intersecting multi-part family, disproving a conjecture of Alon and Katona.