A random version of Sperner's theorem

Let P(n)P(n) denote the power set of [n][n], ordered by inclusion, and let P(n,p)P(n,p) be obtained from P(n)P(n) by selecting elements from P(n)P(n) independently at random with probability p. A classical result of Sperner [12] asserts that every antichain in P(n)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)P(n,p): If pn→∞pn→∞ then, with high probability, the size of the largest antichain in P(n,p)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 pn/log⁡n→∞pn/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.