Persistent Homology and the Upper Box Dimension

We introduce a fractal dimension for a metric space defined in terms of the persistent homology of extremal subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the dimensions coincide for subsets of ${\mathbb{R}}^2$ whose upper box dimension exceeds $\mathrm{1.5.}$ These results are related to extremal questions about the number of persistent homology intervals of a set of $n$ points in a metric space.