Uniform chain decompositions and applications

The Boolean lattice 2[n] is the family of all subsets of [n]={1,…,n} ordered by inclusion, and a chain is a family of pairwise comparable elements of 2[n]. Let s=2n/n⌊n/2⌋, 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 n⌊n/2⌋ 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 Füredi 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 n⌊n/2⌋? We show that the answer is (π8+o(1))2nn. Finally, we discuss how these uniform chain decompositions can be used to optimize and simplify various results in extremal set theory.