Unavoidable subprojections in union-closed set systems of infinite breadth

Authors

Yemon Choi*, Mahya Ghandehari, Hung Le Pham

* Corresponding author

Keywords: breadth, semilattice, subprojection, subquotient, trace of a set system, union-closed set system.

DOI: 10.1016/j.ejc.2021.103311

MSC 2020: Primary 05D10, 06A07. Secondary 06A12.

Status

Published as European J. Combin. 94 (2021), article 103311 (17 pages)

Preprint version available at arXiv 1702.06266v6 (see remarks below regarding version history)

Reviews

[ MR4219308 (pending) | Zbl 07333298 (pending) ]

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.