A random version of Sperner's theorem

Citations of this article
Mendeley users who have this article in their library.


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. © 2014 Elsevier Inc..




Balogh, J., Mycroft, R., & Treglown, A. (2014). A random version of Sperner’s theorem. Journal of Combinatorial Theory. Series A, 128(1), 104–110. https://doi.org/10.1016/j.jcta.2014.08.003

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free