Note on the union-closed sets conjecture
The electronic journal of combinatorics, Tome 24 (2017) no. 3
The union-closed sets conjecture states that if a finite family of sets $\mathcal{A} \neq \{\varnothing\}$ 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.
DOI :
10.37236/6989
Classification :
05D05
Mots-clés : Gowers' polymath project, Reimer's condition
Mots-clés : Gowers' polymath project, Reimer's condition
Affiliations des auteurs :
Abigail Raz 1
@article{10_37236_6989,
author = {Abigail Raz},
title = {Note on the union-closed sets conjecture},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {3},
doi = {10.37236/6989},
zbl = {1369.05198},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6989/}
}
Abigail Raz. Note on the union-closed sets conjecture. The electronic journal of combinatorics, Tome 24 (2017) no. 3. doi: 10.37236/6989
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