Unavoidable subprojections in union-closed set systems of infinite breadth

Choi, Yemon and Ghandehari, Mahya and Pham, Hung Le (2021) Unavoidable subprojections in union-closed set systems of infinite breadth. European Journal of Combinatorics, 94: 103311. ISSN 0195-6698

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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.