Dimension of CPT posets

Abstract

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 $\Delta$ and radius $r$, then \rogers{$dim(\mathcal{P}) \leq \lg\lg \Delta + (\frac{1}{2} + o(1))\lg\lg\lg \Delta + \lg r + \frac{1}{2} \lg\lg r + \frac{1}{2}\lg \pi + 3$. This bound is asymptotically tight up to an additive factor of $\min(\frac{1}{2}\lg\lg\lg \Delta, \frac{1}{2}\lg\lg r)$. 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.