The dimension of interior levels of the Boolean lattice
For a positive integer \(n\) and integers \(k\) and \(r\), \(0 \leq k < r \leq n\), let \(K\) and \(R\) denote (respectively) the families of all \(k\)-element and all \(r\)-element subsets of \(\{1, 2, \dots, n\}\). The dimension \(d(k, r; n)\) of the containment order \(P(k,r;n)\) on \(K \cup R\) is the least number of linear extensions of \(P(k,r;n)\) such that their intersection is \(P(k,r;n)\). The main results describe \(d(k, r;n)\) for the case \(k \geq 2\): 1. For \(n \geq 5\), \(d(2,n - 2;n) = n - 1\). 2. Let \(n\) and \(j\) be positive integers with \(n \geq 5\), \(2 \leq j \leq j_{n - 1}\) and \(\left \lfloor {n - 1 \over j - 1} \right \rfloor \geq 3\). Then let \(r\) be the unique positive integer such that \(\left \lfloor {n - 1\over j} \right \rfloor + j - 2 \leq r - 1 < \left \lfloor {n - 1 \over j - 1} \right \rfloor + j - 3\). (1) If \(n - 1 \not \equiv 0 \bmod (j - 1)\), then \(d(1, r - 1; n - 1) = n - j = d(2, r;n)\). (2) If \(n - 1 \equiv 0 \bmod (j - 1)\), then \(d(1, r - 1; n - 1) = n - j \leq d(2, r;n) \leq n - j + 1\).
scientific article; zbMATH DE number 683159