Order

, Volume 26, Issue 1, pp 31–48

On the Scope of Averaging for Frankl’s Conjecture

  • Gábor Czédli
  • Miklós Maróti
  • E. Tamás Schmidt
Article

DOI: 10.1007/s11083-008-9105-5

Cite this article as:
Czédli, G., Maróti, M. & Schmidt, E.T. Order (2009) 26: 31. doi:10.1007/s11083-008-9105-5

Abstract

Let \(\mathcal F\) be a union-closed family of subsets of an m-element set A. Let \(n=|{\mathcal F}|\ge 2\). For b ∈ A let w(b) denote the number of sets in \(\mathcal F\) containing b minus the number of sets in \(\mathcal F\) not containing b. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element b ∈ A with w(b) ≥ 0. The present paper deals with the average of the w(b), computed over all b ∈ A. \(\mathcal F\) is said to satisfy the averaged Frankl’s property if this average is non-negative. Although this much stronger property does not hold for all union-closed families, the first author (Czédli, J Comb Theory, Ser A, 2008) verified the averaged Frankl’s property whenever n ≥ 2m − 2m/2 and m ≥ 3. The main result of this paper shows that (1) we cannot replace 2m/2 with the upper integer part of 2m/3, and (2) if Frankl’s conjecture is true (at least for m-element base sets) and \(n\ge 2^m-\lfloor 2^m/3\rfloor\) then the averaged Frankl’s property holds (i.e., 2m/2 can be replaced with the lower integer part of 2m/3). The proof combines elementary facts from combinatorics and lattice theory. The paper is self-contained, and the reader is assumed to be familiar neither with lattices nor with combinatorics.

Keywords

Union-closed setsFrankl’s conjectureLattice

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Gábor Czédli
    • 1
  • Miklós Maróti
    • 1
  • E. Tamás Schmidt
    • 2
  1. 1.Bolyai InstituteUniversity of SzegedSzegedHungary
  2. 2.Mathematical Institute of the Budapest University of Technology and EconomicsBudapestHungary