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arXiv 1802.09326math.COcs.DM
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CPT偏序集的维数

Dimension of CPT posets

Atrayee Majumder, Rogers Mathew, Deepak Rajendraprasad

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摘要：

$X$上的线性序的集合，比如$\mathcal{L}$，称为\emph{realize}是偏序集（或偏序集）$\mathcal{P} = (X, \preceq)$，如果对于任意两个不同的$x,y \in X$，$x \preceq y$当且仅当$x \prec_L y$，$\forall L \in \mathcal{L}$。我们称$\mathcal{L}$为$\mathcal{P}$的\emph{realizer}。$\mathcal{P}$的\emph{dimension}，用$dim(\mathcal{P})$表示，是$\mathcal{P}$的实现者的最小基数。偏序集$\mathcal{P}=(X,\preceq)$的\emph{containment model} $M_{\mathcal{P}}$将每个$x \in X$映射到集合$M_x$，使得对于每个不同的$x,y \in X,\ x \preceq y$当且仅当$M_x \varsubsetneq M_y$。我们将使用集合$(M_x)_{x \in X}$来标识包含模型$M_{\mathcal{P}}$。偏序集$\mathcal{P}=(X,\preceq)$是树中路径的包含序（CPT偏序集），如果它允许包含模型$M_{\mathcal{P}}=(P_x)_{x \in X}$，其中每个$P_x$是树$T$的路径，该树被称为模型的宿主树。证明了如果偏序集$\mathcal{P}$在最大度$Δ$和半径$r$的宿主树$T$中存在CPT模型，则\rogers { $dim(\mathcal{P}) \leq \lg\lg Δ+ (\frac{1}{2} + o(1))\lg\lg\lg Δ+ \lg r + \frac{1}{2} \lg\lg r + \frac{1}{2}\lg π+ 3$ .该界是渐近紧的，直到$\min(\frac{1}{2}\lg\lg\lg Δ, \frac{1}{2}\lg\lg r)$的加性因子。进一步，令$\mathcal{P}(1,2;n)$是由$[n]$的所有$1$ -元素和$2$ -元素子集在“包含”关系下组成的偏序集，并令$dim(1,2;n)$表示其维数。我们的主要定理的证明给出了一个简单的算法来构造一个$\mathcal{P}(1,2;n)$的实现，其基数是最多$\frac{3}{2}$远离最优的一个加性因子。

英文摘要：

A collection of linear orders on $X$, say $\mathcal{L}$, is said to \emph{realize} a partially ordered set (or poset) $\mathcal{P} = (X, \preceq)$ if, for any two distinct $x,y \in X$, $x \preceq y$ if and only if $x \prec_L y$, $\forall L \in \mathcal{L}$. We call $\mathcal{L}$ a \emph{realizer} of $\mathcal{P}$. The \emph{dimension} of $\mathcal{P}$, denoted by $dim(\mathcal{P})$, is the minimum cardinality of a realizer of $\mathcal{P}$. A \emph{containment model} $M_{\mathcal{P}}$ of a poset $\mathcal{P}=(X,\preceq)$ maps every $x \in X$ to a set $M_x$ such that, for every distinct $x,y \in X,\ x \preceq y$ if and only if $M_x \varsubsetneq M_y$. We shall be using the collection $(M_x)_{x \in X}$ to identify the containment model $M_{\mathcal{P}}$. A poset $\mathcal{P}=(X,\preceq)$ is a Containment order of Paths in a Tree (CPT poset), if it admits a containment model $M_{\mathcal{P}}=(P_x)_{x \in X}$ where every $P_x$ is a path of a tree $T$, which is called the host tree of the model. We show that if a poset $\mathcal{P}$ admits a CPT model in a host tree $T$ of maximum degree $Δ$ and radius $r$, then \rogers{$dim(\mathcal{P}) \leq \lg\lg Δ+ (\frac{1}{2} + o(1))\lg\lg\lg Δ+ \lg r + \frac{1}{2} \lg\lg r + \frac{1}{2}\lg π+ 3$. This bound is asymptotically tight up to an additive factor of $\min(\frac{1}{2}\lg\lg\lg Δ, \frac{1}{2}\lg\lg r)$. Further, let $\mathcal{P}(1,2;n)$ be the poset consisting of all the $1$-element and $2$-element subsets of $[n]$ under `containment' relation and let $dim(1,2;n)$ denote its dimension. The proof of our main theorem gives a simple algorithm to construct a realizer for $\mathcal{P}(1,2;n)$ whose cardinality is only an additive factor of at most $\frac{3}{2}$ away from the optimum.

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