Uniform chain decompositions and applications

Abstract

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 = 2 n / 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 ( π 8 + o(1))2 n √ n. Finally, we discuss how these uniform chain decompositions can be used to optimize and simplify various results in extremal set theory.

Forbidden subposet problems in the grid

“…Forbidden subposet problems on the grid and their connection to the case of the Boolean cube was first studied by Tomon [22] and Sudakov, Tomon, Wagner [20]. In [20], the following general framework was introduced.…”

“…Forbidden subposet problems on the grid and their connection to the case of the Boolean cube was first studied by Tomon [22] and Sudakov, Tomon, Wagner [20]. In [20], the following general framework was introduced. We say that a formula is affine, if it is built from variables, the lattice operators ∧ and ∨ (or to avoid confusion, one might prefer to use ∪ and ∩), and parentheses (,) (constants are not allowed, e.g.…”

“…We are going to prove a theorem that is similar in flavour to Theorem 1.1. However the proof of Theorem 1.3 is much simpler than that of Theorem 1.1, as the authors of [20] applied an involved chain partition theorem to obtain their result, while for us a relatively simple averaging argument would suffice. Observe that Corollary 1.2 is an immediate consequence of Theorem 1.3 with α = 1.…”