Given a 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. 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 $\{0,1,\ldots,k\}^n$ also has the centeredness property, provided $n$ is sufficiently large compared to $k$. 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, Scott, and Sudakov by showing that the poset of subspaces of $\mathbb{F}_q^n$ has the centeredness property. Several open questions are also given.

中文翻译：

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最小化可比较对的数量的poset中的家庭

给定一个poset $P$，我们说一个族$\mathcal{F}\subseteq P$ 是居中的，如果它是通过“使集合尽可能靠近中间层”获得的。如果对于任何 $M$，在 $P$ 中大小为 $M$ 的所有族中，居中族包含最少数量的可比对，则称偏序 $P$ 具有居中性。Kleitman 证明了布尔格 $\{0,1\}^n$ 具有中心性。Noel、Scott 和 Sudakov 以及 Balogh 和 Wagner 推测，poset $\{0,1,\ldots,k\}^n$ 也具有中心性，前提是 $n$ 足够大到 $k$。我们证明这个猜想对于所有 $k\geq 2$ 都是错误的，并调查了它所适用的 $M$ 的范围。此外，我们改进了 Noel、Scott、和 Sudakov 通过证明 $\mathbb{F}_q^n$ 的子空间的偏序集具有中心性。还给出了几个开放性问题。