Chain Conditions and Optimal Elements in Generalized Union-Closed Families of Sets
Cory H. Colbert
TL;DR
This work investigates how chain conditions on union-closed families interact with the notion of optimal elements to guarantee abundant elements, extending Frankl's union-closed conjecture to certain infinite contexts. It shows existence and utility of optimal elements under descending chain conditions (DCC) and connects ascending chain conditions (ACC) on the associated family of x-stars to abundance results, including in dimension at most two and in topological spaces with DCC on open sets. The paper introduces a separating reduction and the notion of $\alpha$-tents to broaden the applicability of abundance results beyond strictly union-closed families. It also demonstrates that abundant elements can appear in non-union-closed families, highlighting broader reach of Frankl-type conclusions.
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.