Subsets of posets minimising the number of chains

TL;DR

This paper extends classical extremal set theory results by characterizing subsets of an n-element set that minimize the number of chains of a given length, confirming a long-standing prediction and generalizing Kleitman's findings.

Contribution

It generalizes Kleitman's minimal chain count results from length two to arbitrary lengths, providing a complete characterization of families with minimal chains.

Findings

Minimum number of chains of length k is achieved by sets with sizes close to n/2+1/4.

Characterization of all families with the smallest positive number of such chains.

Confirms Kleitman's prediction for all chain lengths and subset sizes.

Abstract

A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection. Generalising this result, Erd\H{o}s characterised the largest families of subsets of an $n$-element set that do not contain a chain of sets $A_1 \subset \dotsc \subset A_k$ 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 far-reaching extension of Sperner's theorem, Kleitman determined the smallest number of chains of length two 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 fifty 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.