Decompositions and connectivity of matching and chessboard complexes (Q701790)

Given integers \(n\geq r\geq 2\), the \(r\)-hypergraph matching complex \(M_n(r)\) is the simplicial complex on the vertex set \(V_n(r)\) of all \(r\)-element subsets of \([n]:=\{1,2,\dots,n\}\) with faces the subsets of \(V_n(r)\) having pairwise disjoint elements. Given positive integers \(n_1,n_2,\dots,n_r\) with \(r\geq 2\), the \((n_1,n_2,\dots,n_r)\)-chessboard complex \(M_{n_1,n_2,\dots,n_r}\) is the simplicial complex on the vertex set \(V=[n_1]\times [n_2]\times \dots \times [n_r]\) with faces the sets of \(r\)-tuples from \(V\) with no two having a coordinate in common. The author proves new lower bounds for the connectivity degree of the simplicial complexes \(M_n(r)\) and \(M_{n_1,n_2,\dots,n_r}\) by showing that certain skeleta of these complexes are vertex decomposable. In particular the bounds given by \textit{A. Björner, L. Lovász, S. T. Vrećica} and \textit{R. T. Živaljević} [J. Lond. Math. Soc., II. Ser. 49, 25--39 (1994; Zbl 0790.57014)] are improved for \(r\geq 3\).