On orthogonal symmetric chain decompositions (Q2325764)

Summary: The \textit{\(n\)-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 \textit{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 \textit{edge-disjoint} symmetric chain decompositions of the \(n\)-cube for \(n\geq 90\), where edge-disjointness is a slightly weaker notion than orthogonality, improving on a recent result by Gregor, Jäger, Mütze, Sawada, and Wille.