# Unavoidable subprojections in union-closed set systems of infinite breadth

## TL;DR

This paper characterizes union-closed set systems with infinite breadth by showing that at least one of three specific configurations must appear as a subprojection, providing a foundational structural insight.

## Contribution

It introduces the first general structural result for union-closed set systems of infinite breadth, identifying unavoidable subprojections.

## Key findings

- At least one of the three configurations occurs in any infinite breadth system.
- The configurations ${
m max}(E)$, ${
m min}(E)$, and ${
m ort}(E)$ are fundamental to the structure.
- This result advances understanding of the internal structure of infinite breadth set systems.

## Abstract

We consider union-closed set systems with infinite breadth, focusing on three particular configurations ${\mathcal T}_{\rm max}(E)$, ${\mathcal T}_{\rm min}(E)$ and ${\mathcal T}_{\rm ort}(E)$. We show that these three configurations are not isolated examples; in any given union-closed set system of infinite breadth, at least one of these three configurations will occur as a subprojection. This characterizes those union-closed set systems which have infinite breadth, and is the first general structural result for such set systems.