Maximum-size antichains in random set-systems

TL;DR

This paper proves that in a random subset of the power set of an n-element set, the maximum size of an antichain avoiding k-chains is approximately (k-1) times the expected size, confirming a conjecture.

Contribution

It establishes the asymptotic size of the largest k-chain-free subset in a random set-family, confirming a conjecture and providing new probabilistic bounds.

Findings

Maximum size of k-chain-free subset is (k-1 + o(1)) p binomial(n, n/2)

Results hold with high probability as pn approaches infinity

Conjecture by Osthus confirmed independently by other researchers

Abstract

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.