  We show that, for $pn \to \infty$, the largest set in a $p$-random sub-family
of the power set of $\{1, \ldots, n\}$ containing no $k$-chain has size $( k -
1 + o(1) ) p \binom{n}{n/2}$ with high probability. This confirms a conjecture
of Osthus, and has been proved independently by Balogh, Mycroft and Treglown.
