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# Journal of Combinatorics

## Volume 3 (2012)

### Number 4

### An extremal problem for set families generated with the union and symmetric difference operations

Pages: 651 – 668

DOI: http://dx.doi.org/10.4310/JOC.2012.v3.n4.a4

#### Authors

#### Abstract

Let ${\mathcal G}$ be a family of sets and let ${\cup}^{n}{\mathcal G}$ be the family of sets obtained by taking all unions of $k$ sets of ${\mathcal G}$ with $1\leq k\leq n$. We define the $\textit{half-life}$ of ${\mathcal G}$ with respect to the union operation, denoted by $h_{\cup}({\mathcal G})$, to be the smallest integer $n$ such that some $x\in\cup_{A\in{\mathcal G}}A$ appears in at least half of the sets in ${\cup}^{n}{\mathcal G}$. If no such $n$ exists, then we define it as $\infty$. We also define the *half-life* of ${\mathcal G}$ with respect to the symmetric difference operation in a similar fashion and denote it by $h_{\Delta}({\mathcal G})$. In this paper, we establish several bounds for $h_{\cup}({\mathcal G})$ and $h_{\Delta}({\mathcal G})$. As a byproduct, we confirm Fránkl’s union-closed conjecture for some special cases.

#### Keywords

union-closed sets, union-closed conjecture, half-life

#### 2010 Mathematics Subject Classification

05D05

Published 21 February 2013