A combinatorial idea of Gromov is to assign to each metric space X the
collection of all distance matrices corresponding to all possible n-tuples of
points in X. Given a filtration the functor F on finite
metric spaces we consider the set of all
possible F-persistence diagrams generated by metric
subsets of X of cardinality n. For a class of filtration functors
which we call compatible, the answer is positive, and these admit stability results in the Gromov-Hausdorff sense.