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On Dedekind's problem, a sparse version of Sperner's theorem, and antichains of a given size in the Boolean lattice
Matthew Jenssen, Alexandru Malekshahian, Jinyoung Park
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Dedekind's problem, dating back to 1897, asks for the total number ψ(n) of antichains contained in the Boolean lattice Bₙ on n elements. We study Dedekind's problem using a recently developed method based on the cluster expansion from statistical physics and as a result, obtain several new results on the number and typical structure of antichains in Bₙ. We obtain detailed estimates for both ψ(n) and the number of antichains of size βn⌊n/2 ⌋ for any fixed β>0. We also establish a sparse version of Sperner's theorem: we determine the sharp threshold and scaling window for the property that almost every antichain of size m is contained in a middle layer of Bₙ.
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