Partition‐free families of sets. Issue 2 (25th February 2019)

• Record Type: Journal Article
• Authors: Frankl, Peter; Kupavskii, Andrey
• Is Part Of: Proceedings of the London Mathematical Society. Volume 119:Issue 2(2019)
• Page Start: 440
• Page End: 468
• Publication Date: 2019-02-25
• Subjects: 05D05 -- 05D15 (primary); Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12236
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract: Let m ( n ) denote the maximum size of a family of subsets which does not contain two disjoint sets along with their union. In 1968, Kleitman proved that m ( n ) = n m + 1 + ⋯ + n 2 m + 1 if n = 3 m + 1 . Confirming the conjecture of Kleitman, we establish the same equality for the cases n = 3 m and n = 3 m + 2, and also determine all extremal families. Unlike the case n = 3 m + 1, the extremal families are not unique. This is a plausible reason behind the relative difficulty of our proofs. We completely settle the case of several families as well.