Families in posets minimizing the number of comparable pairs

• Published in: Journal of graph theory Vol. 95; no. 4; pp. 655 - 676
• Main Authors: Balogh, József, Petříčková, Šárka, Wagner, Adam Zsolt
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.12.2020
• Subjects: Boolean algebra; Boolean lattice; comparable pair; Kleitman's theorem; Noel, Scott; poset; Set theory; Sperner's theorem; Subspaces
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.22604

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.