More on 3-sets in union-closed families: The End is in Sight.
by
Theresa Vaughan
UNC-Greensboro

A union-closed (UC) family is a finite family of finite sets which is closed under unions. (These can be identified, usually in more than one way, with families of positive integers which are closed under least common multiples.) The Frankl Conjecture says that for a UC family, there must be some element which is contained in at least half the members of the family. I have shown that given a UC family with union {1, 2, ..., n}, if the family fails the conjecture, then it cannot contain more than 2n/3 3-sets. With more work, presently underway, I will be able to show such a family contains no more than n/2 3-sets.

Date received: March 31, 2005