Extreme k-families

AbstractLet P be a poset. A subset A of P is a k-family iff A contains no (k + 1)-element chain. For i ⩾ 1, let Ai be the set of elements of A at depth i − 1 in A. The k-families of P can be ordered by defining A ⩽ B iff, for all i, Ai is included in the order ideal generated by Bi. This paper examines minimal r-element k-families, defined as k-families A such that |A| = r and for every B < A, |B| < r. Minimal k-families are related to maximal r-antichains and an operation called Sperner closure, which have been used to obtain extremal results for families of sets with width restrictions. Let Mk,r be the set of minimal r-element k-families and let Mk = ∪r ≥ 0 Mk,r. It is shown that Mk is a join-subsemilattice by the lattice Ak of k-families. Mk is a lower semimodular lattice, where the rth rank is given by Mk,r. If wk is the maximum size of a k-family, then |Mk,r| ⩽ (wrk)and |∪Mk| ⩽ Σi = 1wk ⌈i/k⌉. Let D(A) = max{|B| − |A| | B is a k-family and B ⩽ A}. For k-families A and B, D(A v B) ⩽ D(A) + D(B). This result shows that {A | D(A) = 0} is also a join-subsemilattice of Ak