Abstract: There has been recent interest in graph-based nearest neighbor search methods, many of which are centered on the construction (approximately) "navigable" graphs over high-dimensional point sets. A graph is navigable if greedy routing successfully routes from any starting node to any target node. The complete graph is navigable for any point set, but the important question for applications is if sparser graphs can be constructed. While this question is well understood in low-dimensions, we establish some of the first upper and lower bounds for high-dimensional point sets. First, we give a simple and efficient way to construct a navigable graph with average degree $O(\sqrt{n \log n })$ for any set of $n$ points, in any dimension, for any distance function. We compliment this result with a nearly matching lower bound: even under the Euclidean metric in $O(\log n)$ dimensions, a random point set has no navigable graph with average degree $O(n^{\alpha})$ for any $\alpha < 1/2$. Our lower bound relies on sharp anti-concentration bounds for binomial random variables, which we use to show that the {near-neighborhoods} of a set of random points do not overlap significantly, forcing any navigable graph to have many edges.