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 ${\mathcal{T}}_{\max }(E)$, ${\mathcal{T}}_{\min }(E)$ and ${\mathcal{T}}_{\mathrm{ort}}(E)$. 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.

v5: AMS LaTeX, 21 pages, 1 figure (tikz); uses the stmaryrd fonts. Incorporates referees' feedback. v6: updated to include two minor corrections found after acceptance, which are both included in the final published version. To appear in European J. Combin

References in corpus (3)

• Vapnik-Chervonenkis density in some theories without the independence property, II
• Approximately multiplicative maps from weighted semilattice algebras
• Stability of characters and filters for weighted semilattices

Cited by in corpus (1)

• Constructing non-AMNM weighted convolution algebras for every semilattice of infinite breadth