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  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  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