Families Implying the Frankl Conjecture

Abstract

A union closed (UC) family A is a finite family of sets such that the union of any two sets in A is also in A. Peter Frankl conjectured that for every union closed family A, there exists some x contained in at least half the members of A. An FC-family is a UC family B such that for every UC family A, if B⊆A, then A satisfies the Frankl conjecture. We construct several such families, involving five, six, and seven elements.