Subsets of posets minimising the number of chains

A well-known theorem of Sperner describes the largest collections of subsets of an n-element set, none of which contain another set from the collection. Generalising this result, Erdős characterised the largest families of subsets of an n-element set that do not contain a chain of sets A₁ ⊂ ・・ ・ ⊂ A_k of an arbitrary length k. The extremal families contain all subsets whose cardinalities belong to an interval of length k − 1 centred at n/2. In a farreaching extension of Sperner’s theorem, Kleitman determined the smallest number of chains of length 2 that have to appear in a collection of a given number a of subsets of an n-element set. For every a, this minimum is achieved by the collection comprising a sets whose cardinalities are as close to n/2+1/4 as possible. We show that the same is true about chains of an arbitrary length k, for all a and n, confirming the prediction Kleitman made 50 years ago. We also characterise all families of a subsets with the smallest number of chains of length k for all a for which this smallest number is positive. Our argument is inspired by an elegant probabilistic lemma from a recent paper of Noel, Scott, and Sudakov, which in turn can be traced back to Lubell’s proof of Sperner’s theorem.