Maximum union-free subfamilies
Name
Fox_Maximum union.pdf
Size
159.66 KB
Format
Adobe PDF
Checksum (MD5)
4999f64e253598372891db735b092d8a
Author(s)
Fox, Jacob
Lee, Choongbum
Sudakov, Benny
Date Issued
February 2012
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
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s11856-012-0017-0
