Maurício Collares Neto, Robert Morris
Published 2014-04-21, updated 2015-11-12Version 3
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.
Comments: 14 pages, added important observation to the Introduction
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