Unavoidable subprojections in union-closed set systems of infinite breadth

Abstract: We consider union-closed set systems with infinite breadth, focusing on three particular configurations ${\mathcal T}<em>{\rm max}(E)$, ${\mathcal T}</em>{\rm min}(E)$ 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.