Maximum-size antichains in random set-systems

Abstract

We show that, for $pn\to \infty$, the largest set in a $p$-random sub-family of the power set of $\{1,\dots ,n\}$ containing no $k$-chain has size $(k-1+o(1))p\left(\genfrac{}{}{0ex}{}{n}{n/2}\right)$ with high probability. This confirms a conjecture of Osthus, and has been proved independently by Balogh, Mycroft and Treglown.

14 pages, added important observation to the Introduction