  Recent works of Alon-Shapira and R\"odl-Schacht have demonstrated that every
hereditary property of undirected graphs or hypergraphs is testable with
one-sided error; informally, this means that if a graph or hypergraph satisfies
that property "locally" with sufficiently high probability, then it can be
perturbed (or "repaired") into a graph or hypergraph which satisfies that
property "globally".
  In this paper we make some refinements to these results, some of which may be
surprising. In the positive direction, we strengthen the results to cover
hereditary properties of multiple directed polychromatic graphs and
hypergraphs. In the case of undirected graphs, we extend the result to
continuous graphs on probability spaces, and show that the repair algorithm is
"local" in the sense that it only depends on a bounded amount of data; in
particular, the graph can be repaired in a time linear in the number of edges.
We also show that local repairability also holds for monotone or partite
hypergraph properties (this latter result is also implicitly in work of
Ishigami). In the negative direction, we show that local repairability breaks
down for directed graphs, or for undirected 3-uniform hypergraphs. The reason
for this contrast in behavior stems from (the limitations of) Ramsey theory.
