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arXiv · 1704.07022

Note on the union-closed sets conjecture

Abigail Raz
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Abstract

The union-closed sets conjecture states that if a family of sets $\mathcal{A} \neq \{\emptyset\}$ is union-closed, then there is an element which belongs to at least half the sets in $\mathcal{A}$. In 2001, D. Reimer showed that the average set size of a union-closed family, $\mathcal{A}$, is at least $\frac{1}{2} \log_2 |\mathcal{A}|$. 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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Abigail Raz. 2017-04-24. Note on the union-closed sets conjecture. https://arxiv.org/abs/1704.07022

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