Several Minimality Concepts Related to Frankl’s Conjecture

We study Frankl’s conjecture, also known as the union-closed set conjecture, which is a long-standing open problem in combinatorics. We consider the conjecture in the form of families that are closed under intersection in this paper. The intersection form of the conjecture states that, for any finite family that is closed under intersection, there exists an element in the ground set that belongs to at most half of the members of the family. We introduce a (pre)order on all the families closed under intersection, called the family order, and define several operations along the family order. These operations define several minimal families that are kinds of relaxations of minimal counterexamples to Frankl’s conjecture. As an application, we show that an intersection-closed family that has a 2-transversal will not be a minimal counterexample to Frankl’s conjecture.