  We develop a new technique for establishing quantitative propagation of chaos
for systems of interacting particles. Using this technique we prove propagation
of chaos for diffusing particles whose interaction kernel is merely H\"older
continuous, even at long ranges. Moreover, we do not require specially prepared
initial data. On the way, we establish a law of large numbers for SDEs that
holds over a class of vector fields simultaneously. The proofs bring together
ideas from empirical process theory and stochastic flows.
