Note on the union-closed sets conjecture

2017/04/24 by Raz, Abigail
#05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1704.07022

Abstract

The union-closed sets conjecture states that if a family of sets A ≠ \∅\ 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.

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