# Orthogonal Symmetric Chain Decompositions of Hypercubes

## TL;DR

This paper advances the understanding of hypercube decompositions by constructing three orthogonal chain decompositions for large hypercubes, introducing the concept of almost orthogonal symmetric chain decompositions, and providing explicit examples and conditions for their existence.

## Contribution

It presents the first non-trivial progress on Shearer and Kleitman's conjecture by constructing three orthogonal chain decompositions for large hypercubes and introduces the novel notion of almost orthogonal symmetric chain decompositions.

## Key findings

- Constructed three orthogonal chain decompositions of large hypercubes.
- Explicitly described three such decompositions for Q_5 and Q_7.
- Provided conditions for decomposing products of hypercubes into almost orthogonal symmetric chain decompositions.

## Abstract

In 1979, Shearer and Kleitman conjectured that there exist $\lfloor n/2 \rfloor+1$ orthogonal chain decompositions of the hypercube $Q_n$, and constructed two orthogonal chain decompositions. In this paper, we make the first non-trivial progress on this conjecture since by constructing three orthogonal chain decompositions of $Q_n$ for $n$ large enough. To do this, we introduce the notion of "almost orthogonal symmetric chain decompositions". We explicitly describe three such decompositions of $Q_5$ and $Q_7$, and describe conditions which allow us to decompose products of hypercubes into $k$ almost orthogonal symmetric chain decompositions given such decompositions of the original hypercubes.