Families in posets minimizing the number of comparable pairs. (English) Zbl 1525.05183
Summary: Given a graded poset \(P\) we say a family \(\mathcal{F} \subseteq P\) 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. D. Kleitman [Theory of Graphs, Proc. Colloq. Tihany, Hungary 1966, 215–218 (1968; Zbl 0159.30401)] showed that the Boolean lattice \(\{ 0,1\}^n\) has the centeredness property. It was conjectured by J. A. Noel et al. [J. Comb. Theory, Ser. A 154, 247–284 (2018; Zbl 1373.05197)], and by J. Balogh and A. Z. Wagner [Adv. Math. 330, 229–252 (2018; Zbl 1390.05231)], that the poset \(\{ 0,1,\ldots, k\}^n\) also has the centeredness property, provided \(n\) is sufficiently large compared with \(k\). We show that this conjecture is false for all \(k \geq 2\) and investigate the range of \(M\) for which it holds. Further, we improve a result of Noel et al. [loc. cit.] by showing that the poset of subspaces of \(\mathbb{F}_q^n\) has the centeredness property. Several open questions are also given.
MSC:
| 05D05 | Extremal set theory |
References:
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