Families in posets minimizing the number of comparable pairs

Given a graded poset P we say a family F ⊆ P is centered if it is obtained by ‘taking sets as close to the middle layer as possible.’ A poset P is said to have the centeredness property if for any M , among all families of size M in P , centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice{ 0 , 1 } n has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset{ 0 , 1 , … , k } n also has the centeredness property, provided n is sufficiently large compared with k . We show that this conjecture is false for all k ≥ 2 and investigate the range of M for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of F q n has the centeredness property. Several open questions are also given.