Abstract
The n-cube is the poset obtained by ordering all subsets of {1, ..., n} by inclusion, and it can be partitioned into ((n)(left perpendicular n/2 right perpendicular)) 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 Kleit-man conjectured in 1979 that the n-cube has left perpendicular n/2 right perpendicular+ 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 >= 24.
In this paper, we construct four pairwise orthogonal chain decompositions of the n-cube for n >= 60. We also construct five pairwise edge-disjoint symmetric chain decompositions of the n-cube for n >= 90, where edge-disjointness is a slightly weaker notion than orthogonality, improving on a recent result by Gregor, Jager, Mutze, Sawada, and Wille.