A random version of Sperner's theorem

@article{Balogh2014ARV,
  title={A random version of Sperner's theorem},
  author={J{\'o}zsef Balogh and Richard Mycroft and Andrew Treglown},
  journal={J. Comb. Theory, Ser. A},
  year={2014},
  volume={128},
  pages={104-110}
}
Let P(n) denote the power set of [n], ordered by inclusion, and let P(n, p) be obtained from P(n) by selecting elements from P(n) independently at random with probability p. A classical result of Sperner [12] asserts that every antichain in 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): If pn → ∞ then, with high probability, the size of the largest antichain in P(n, p) is at most (1+ o(1))p ( n ⌊n/2⌋ ) . This solves a… CONTINUE READING

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