On orthogonal symmetric chain decompositions

TL;DR

This paper advances the understanding of orthogonal chain decompositions of the n-cube by constructing four such decompositions for large n and five edge-disjoint decompositions for even larger n, improving previous results.

Contribution

The paper constructs four pairwise orthogonal chain decompositions of the n-cube for n≥60 and five edge-disjoint decompositions for n≥90, surpassing prior known bounds.

Findings

Constructed four orthogonal chain decompositions for n≥60.

Constructed five edge-disjoint chain decompositions for n≥90.

Improved bounds on the number of such decompositions in the n-cube.

Abstract

The $n$-cube is the poset obtained by ordering all subsets of $\{1,\ldots,n\}$ by inclusion, and it can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains, which is the minimum possible number. Two such decompositions of the $n$-cube are called orthogonal if any two chains of the decompositions share at most a single element. Shearer and Kleitman conjectured in 1979 that the $n$-cube has $\lfloor n/2\rfloor+1$ pairwise orthogonal decompositions into the minimum number of chains, and they constructed two such decompositions. Spink recently improved this by showing that the $n$-cube has three pairwise orthogonal chain decompositions for $n\geq 24$. In this paper, we construct four pairwise orthogonal chain decompositions of the $n$-cube for $n\geq 60$. We also construct five pairwise edge-disjoint chain decompositions of the $n$-cube for $n\geq 90$, where edge-disjointness is a slightly weaker notion than orthogonality.