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

中文翻译：

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CPT 姿势的尺寸

X 上的线性阶集合，比如 L ${\mathscr{L}}$ ，据说实现了偏序集（或poset）P = ( X , ≼ ) $\mathcal {P} = (X, \ preceq )$ 如果，对于任意两个不同的 x ， y ∈ X ， x ≼ y 当且仅当 x ≺ L y ，∀ L ∈ L $\forall L \in {\mathscr{L}}$ 。我们称 L ${\mathscr{L}}$ 是 P $\mathcal {P}$ 的实现者。P $\mathcal {P}$ 的维度，用dim ( P ) $dim(\mathcal {P})$ 表示，是P $\mathcal {P}$ 的实现者的最小基数。一个偏序集 P = ( X , ≼ ) $\mathcal {P}=(X,\preceq )$ 的包含模型 MP $M_{\mathcal {P}}$ 将每个 x ∈ X 映射到一个集合 M x 使得, 对于每个不同的 x , y ∈ X , x ≼ y 当且仅当 M x ⫋ M y $M_{x} \varsubsetneq M_{y}$ 。我们将使用集合 ( M x ) x ∈ X 来识别包含模型 MP $M_{\mathcal {P}}$ 。一个偏序 P = ( X , ≼ ) $\mathcal {P}=(X, \preceq )$ 是树中路径的包含顺序（CPT poset），如果它承认包含模型 MP = ( P x ) x ∈ X $M_{\mathcal {P}}=(P_{x})_ {x \in X}$ 其中每个 P x 是树 T 的路径，称为模型的宿主树。我们证明，如果偏序集 P $\mathcal {P}$ 在最大度为 Δ 和半径为 r 的宿主树 T 中接纳 CPT 模型，则 dim ( P ) ≤ lg lg Δ + ( 1 2 + o ( 1 ) ) lg lg lg Δ + lg r + 1 2 lg lg r + 1 2 lg π + 3 $dim(\mathcal {P}) \leq \lg \lg {\Delta } + (\frac {1}{2} + o(1))\lg \lg \lg {\Delta } + \lg r + \frac {1}{2} \lg \lg r + \frac {1}{2}\lg \pi + 3$ 。这个界限是渐近紧的，直到一个加性因子 min ( 1 2 lg lg lg Δ , 1 2 lg lg r ) $\min \limits (\frac {1}{2}\lg \lg \lg {\Delta } , \frac {1}{2}\lg \lg r)$ 。进一步，令 P ( 1 , 2 ; n ) $\mathcal {P}(1,2; n)$ 是由 [ n ] 在“包含”关系下的所有 1 元素和 2 元素子集组成的偏序集，并让 dim (1,2; n ) 表示其维度。我们的主要定理的证明给出了一个简单的算法来构造 P ( 1 , 2 ; n ) $\mathcal {P}(1,2;n)$ 的实现器，其基数只是最多 3 2 $ 的加法因子\frac {3}{2}$ 远离最佳。