Note on the Union-Closed Sets Conjecture

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 preceding bound. Here, answering a question raised in the context of T. Gowers’ polymath project on the union-closed sets conjecture, we show that Reimer’s condition alone is not enough to imply that there is an element in at least half the sets.