Small Sets in Union-Closed Families

Abstract

Our aim in this note is to show that, for any ǫ > 0, there exists a union-closed family F with (unique) smallest set S such that no element of S belongs to more than a fraction ǫ of the sets in F . More precisely, we give an example of a union-closed family with smallest set of size k such that no element of this set belongs to more than a fraction (1 + o(1)) log 2 k 2k of the sets in F . We also give explicit examples of union-closed families containing 'small' sets for which we have been unable to verify the Union-Closed Conjecture.

Notes on the Union Closed Sets Conjecture

“…For more details on the development of the conjecture and what is known about it see the survey [7]. In the recent years [1,2,6,12,18,21,28,29,30] have further been published investigating the conjecture with respect to one aspect or another. In this context, the collaborative effort in [16] should also be mentioned.…”