Maximum union-free subfamilies

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 √4m + 1 − 1 and that this bound is tight. This solves Moser’s problem and proves a conjecture of Erdős 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, Füredi, Kantor, Kim and Patkos. ∗ Research supported by a Simons Fellowship and NSF grant DMS-1069197. ∗∗ Research supported in part by a Samsung Scholarship. † Research supported in part by NSF grant DMS-1101185, NSF CAREER award DMS-0812005 and by a USA–Israeli BSF grant. Received January 10, 2011