Maximum union-free subfamilies

An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? 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 4m + 1 - 1 and 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 a + 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, F

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Jacob Fox, Choongbum Lee, Benny Sudakov
• Keywords: CORR 2010, Distinct Sets, Education, Multiplicative Constant Factor, Union-free Subfamily