Uniform chain decompositions and applications

TL;DR

This paper proves a conjecture on partitioning the Boolean lattice into nearly equal-sized chains, introduces a probabilistic and graph container method approach, and applies these results to extremal set theory problems.

Contribution

It provides an asymptotic partition of the Boolean lattice into chains of nearly equal size, confirming Furedi's 1985 conjecture, and develops new probabilistic and weighted graph container techniques.

Findings

Partition of $2^{[n]}$ into $inom{n}{loor{n/2}}$ chains with nearly equal size

Answer to a Kalai-type extremal problem with minimal forbidden pairs

Development of a weighted graph container method for probabilistic analysis

Abstract

The Boolean lattice $2^{[n]}$ is the family of all subsets of $[n]=\{1,\dots,n\}$ ordered by inclusion, and a chain is a family of pairwise comparable elements of $2^{[n]}$. Let $s=2^{n}/\binom{n}{\lfloor n/2\rfloor}$, which is the average size of a chain in a minimal chain decomposition of $2^{[n]}$. We prove that $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that all but at most $o(1)$ proportion of the chains have size $s(1+o(1))$. This asymptotically proves a conjecture of F\"uredi from 1985. Our proof is based on probabilistic arguments. To analyze our random partition we develop a weighted variant of the graph container method. Using this result, we also answer a Kalai-type question raised recently by Das, Lamaison and Tran. What is the minimum number of forbidden comparable pairs forcing that the largest subfamily of $2^{[n]}$ not containing any of them has size at most $\binom{n}{\lfloor n/2\rfloor}$? We show that the answer is $(\sqrt{\frac{\pi}{8}}+o(1))2^{n}\sqrt{n}$. Finally, we discuss how these uniform chain decompositions can be used to optimize and simplify various results in extremal set theory.