Chain Conditions and Optimal Elements in Generalized Union-Closed Families of Sets

TL;DR

This paper explores conditions under which the union-closed sets conjecture holds, extending its validity to certain infinite families and topological spaces using chain conditions and optimal elements.

Contribution

It introduces chain conditions and optimal elements to prove the conjecture for specific infinite and topological families, broadening the scope of known results.

Findings

The conjecture holds for families with chains of size at most three.

It is valid for certain topological spaces with the descending chain condition.

Identifies classes of families satisfying the conjecture without union-closure.

Abstract

The union-closed sets conjecture (sometimes referred to as Frankl's conjecture) states that every finite, nontrivial union-closed family of sets has an element that is in at least half of its members. Although the conjecture is known to be false in the infinite setting, we show that many interesting results can still be recovered by imposing suitable chain conditions and considering carefully chosen elements called optimal elements. We use these elements to show that the union-closed conjecture holds for both finite and infinite union-closed families such that the cardinality of any chain of sets is at most three. We also show that the conjecture holds for all nontrivial topological spaces satisfying the descending chain condition on its open sets. Notably, none of those arguments depend on the cardinality of the underlying family or its universe. Finally, we provide an interesting class of families that satisfy the conclusion of the conjecture but are not necessarily union-closed.