Packing and covering k-chain free subsets in Boolean lattices

Abstract

a b s t r a c tLet P be a finite poset. A subset of P is called k-element chain free if it contains no k-element chain. Let H (B n ) be the hypergraph whose vertices are the points of the Boolean lattice B n , and whose edges are inclusionwise maximal k-chain free subsets of B n . We investigate the covering number τ k (B n ) of H, i.e. the least number of points intersecting every maximal k-chain free subset of B n , and the packing (matching) number ν k (B n ) of H , i.e. the greatest number of pairwise disjoint maximal k-chain free subsets in B n . Using counting arguments, exponential bounds for τ k (B n ) and ν k (B n ) are given.