A random version of Sperner's theorem

József Balogh, Richard Mycroft, Andrew Treglown

Research output: Contribution to journalArticlepeer-review


Let P(n) denote the power set of [. n], ordered by inclusion, and let P(n,p) be obtained from P(n) by selecting elements from P(n) independently at random with probability p. A classical result of Sperner [12] asserts that every antichain in P(n) has size at most that of the middle layer, (n⌊n/2⌋). In this note we prove an analogous result for P(n,p): If p n→ ∞ then, with high probability, the size of the largest antichain in P(n,p) is at most (1+o(1))p(n⌊n/2⌋). This solves a conjecture of Osthus [9] who proved the result in the case when p n/log n→ ∞ Our condition on p is best-possible. In fact, we prove a more general result giving an upper bound on the size of the largest antichain for a wider range of values of p.

Original languageEnglish (US)
Pages (from-to)104-110
Number of pages7
JournalJournal of Combinatorial Theory. Series A
Issue number1
StatePublished - Nov 2014


  • Antichain
  • Boolean lattice
  • Container method

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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