@article{DaeubelJaegerMuetzeScheucher2019b,
Title = {On orthogonal symmetric chain decompositions},
Author = {D{\"a}ubel, Karl and J\"{a}ger, Sven and M{\"u}tze, Torsten and Scheucher, Manfred},
Pages = {P3.64},
Year = {2019},
Issn = {1077-8926},
Journal = {Electronic Journal of Combinatorics},
Volume = {26},
Number = {3},
Month = {09},
Note = {Full version},
Abstract = {The n-cube is the poset obtained by ordering all subsets of {1,…,n} by inclusion, and it can be partitioned into n choose ⌊n/2⌋ 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 ⌊n/2⌋+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, Jäger, Mütze, Sawada, and Wille.},
Url2 = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i3p64}
}