Packing and covering -chain free subsets in Boolean lattices

Let 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.