The order dimension of two levels of the Boolean lattices
The Boolean lattice of all subsets of a set having \(n\) elements is denoted by \(B_ n\). If \(0\leq s< t\leq n\), then \(B_ n(s,t)\) is the subset of \(B_ n\) consisting of all elements of \(B_ n\) which are sets with \(s\) or with \(t\) elements; the ordering on \(B_ n(s,t)\) is the restriction of that on \(B_ n\). The dimension of a partially ordered set \(P\) is the minimum positive integer \(d\) such that \(P\) can be embedded into the \(d\)th Cartesian power of the (naturally ordered) set of all real numbers. The dimension of \(B_ n(s,t)\) is denoted by \(\dim(n; s,t)\). The paper gives a survey of results concerning \(\dim(n; s,t)\). Then two inequalities are stated and the main theorem is proved: For \(k\geq 3\) and \(n> 250 k^ 3\), we have \(\dim(n; k,n- k)= n- 2\).
scientific article; zbMATH DE number 683158