We give a new proof of theorem Lo Faro concerning minimal counterexamples to the union-closed conjecture, and we prove some results “ε-version” maximal frequency in families.

The union-closed sets conjecture states that if a finite family of sets A 6 = {∅} is union-closed, then there is an element which belongs to at least half the sets in A. In 2001, D. Reimer showed that the average set size of a union-closed family, A, is at least 1 2 log2 |A|. In order to do so, he showed that all union-closed families satisfy a particular condition, which in turn implies the pr...

 In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...