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Maximum union-free subfamilies

Author(s)
Fox, Jacob; Lee, Choongbum; Sudakov, Benny
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Abstract
An old problem of Moser asks: what is the size of the largest union-free subfamily that one can guarantee in every family of m sets? A family of sets is called union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. We show that every family of m sets contains a union-free subfamily of size at least [[sqrt 4m+1]] - 1 that this bound is tight. This solves Moser’s problem and proves a conjecture of Erdos and Shelah from 1972. More generally, a family of sets is a-union-free if there are no alpha + 1 distinct sets in the family such that one of them is equal to the union of a others. We determine up to an absolute multiplicative constant factor the size of the largest guaranteed a-union-free subfamily of a family of m sets. Our result verifies in a strong form a conjecture of Barat, Füredi, Kantor, Kim and Patkos.
Date issued
2012-02
URI
http://hdl.handle.net/1721.1/70972
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Israel Journal of Mathematics
Publisher
Springer Science + Business Media B.V.
Citation
Fox, Jacob, Choongbum Lee, and Benny Sudakov. “Maximum Union-free Subfamilies.” Israel Journal of Mathematics (2012): 1-13. Web.
Version: Author's final manuscript
ISSN
0021-2172
1565-8511

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