Threshold Functions for Random Graphs on a Line Segment

We look at a model of random graphs suggested by Gilbert: given an integer $n$ and $\delta > 0$, scatter $n$ vertices independently and uniformly on a metric space, and then add edges connecting pairs of vertices of distance less than $\delta$ apart. We consider the asymptotics when the metric space is the interval [0, 1], and $\delta = \delta(n)$ is a function of $n$, for $n \to \infty$. We prove that every upwards closed property of (ordered) graphs has at least a weak threshold in this model on this metric space. (But we do find a metric space on which some upwards closed properties do not even have weak thresholds in this model.) We also prove that every upwards closed property with a threshold much above connectivity's threshold has a strong threshold. (But we also find a sequence of upwards closed properties with lower thresholds that are strictly weak.)