Minimum saturated families of sets

Matija Bucić, Shoham Letzter, Benny Sudakov, Tuan Tran

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We call a family F of subsets of [n] s-saturated if it contains no s pairwise disjoint sets, and moreover no set can be added to F while preserving this property (here [n] = {1, . . . , n}). More than 40 years ago, Erdős and Kleitman conjectured that an s -saturated family of subsets of [n] has size at least (1 − 2−(s−1))2n. It is easy to show that every s -saturated family has size at least 1/2.2n, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of (1/2 + ε)2n, for some fixed ε > 0, seems difficult. In this note, we prove such a result, showing that every s -saturated family of subsets of [n] has size at least (1 − 1/s)2n. This lower bound is a consequence of a multipartite version of the problem, in which we seek a lower bound on |F1| + · · · + |Fs| where F1, . . . ,Fs are families of subsets of [n], such that there are no s pairwise disjoint sets, one from each family Fi, and furthermore no set can be added to any of the families while preserving this property. We show that |F1| + · · · + |Fs| ≥ (s − 1) · 2n, which is tight, for example, by taking F1 to be empty, and letting the remaining families be the families of all subsets of [n].

Original languageEnglish (US)
Pages (from-to)725-732
Number of pages8
JournalBulletin of the London Mathematical Society
Volume50
Issue number4
DOIs
StatePublished - Aug 2018
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • 05D05 (primary)
  • 60C05 (secondary)

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