  The page number of a directed acyclic graph $G$ is the minimum $k$ for which
there is a topological ordering of $G$ and a $k$-coloring of the edges such
that no two edges of the same color cross, i.e., have alternating endpoints
along the topological ordering. We address the long-standing open problem
asking for the largest page number among all upward planar graphs. We improve
the best known lower bound to $5$ and present the first asymptotic improvement
over the trivial $O(n)$ upper bound, where $n$ denotes the number of vertices
in $G$. Specifically, we first prove that the page number of every upward
planar graph is bounded in terms of its width, as well as its height. We then
combine both approaches to show that every $n$-vertex upward planar graph has
page number $O(n^{2/3} \log(n)^{2/3})$.
