Our aim in this note is to show that, for any $\epsilon>0$, there exists a union-closed family $\mathcal F$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $\epsilon$ of the sets in $\mathcal F$. 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 $(1+o(1))\frac{\log_2 k}{2k}$ of the sets in $\mathcal F$. We also give explicit examples of union-closed families containing 'small' sets for which we have been unable to verify the Union-Closed Conjecture.
@article{10_37236_11004,
author = {David Ellis and Maria-Romina Ivan and Imre Leader},
title = {Small sets in union-closed families},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/11004},
zbl = {1506.05204},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11004/}
}
TY - JOUR
AU - David Ellis
AU - Maria-Romina Ivan
AU - Imre Leader
TI - Small sets in union-closed families
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11004/
DO - 10.37236/11004
ID - 10_37236_11004
ER -
%0 Journal Article
%A David Ellis
%A Maria-Romina Ivan
%A Imre Leader
%T Small sets in union-closed families
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11004/
%R 10.37236/11004
%F 10_37236_11004
David Ellis; Maria-Romina Ivan; Imre Leader. Small sets in union-closed families. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11004
