Families implying the Frankl conjecture (Q1864599)

A union-closed (UC) family \({\mathcal A}\) is a finite collection of sets such that the union of any two members of \({\mathcal A}\) is also a member of \({\mathcal A}\). It was conjectured by \textit{P. Frankl} [Discrete Math. 26, 111-118 (1979; Zbl 0397.05004)] that, for any such \({\mathcal A}\), there must be an element \(x\) in at least half of the sets of \({\mathcal A}\). This conjecture is still open. Define a \(\text{FC}(n)\)-family to be a collection \({\mathcal B}\) of sets, \(|S|= n\) where \(S= \bigcup_{B\in{\mathcal B}} B\), such that, if \({\mathcal A}\) is any UC-family containing \({\mathcal B}\), then one of the elements of \(S\) is contained in at least half of the members of \({\mathcal A}\). Such a \({\mathcal B}\) which does not contain any \(\text{FC}(m)\)-family for any \(m< n\) is called a proper \(\text{FC}(n)\)-family. The author obtains proper \(\text{FC}(n)\)-families for \(n= 5,6\) and \(7\), making use of results of \textit{B. Poonen} [J. Comb. Theory, Ser. A 59, 253-268 (1992; Zbl 0758.05096)].