  Consider several independent Poisson point processes on R^d, each with a
different colour and perhaps a different intensity, and suppose we are given a
set of allowed family types, each of which is a multiset of colours such as
red-blue or red-red-green. We study translation-invariant schemes for
partitioning the points into families of allowed types. This generalizes the
1-colour and 2-colour matching schemes studied previously (where the sets of
allowed family types are the singletons {red-red} and {red-blue} respectively).
We characterize when such a scheme exists, as well as the optimal tail
behaviour of a typical family diameter. The latter has two different regimes
that are analogous to the 1-colour and 2-colour cases, and correspond to the
intensity vector lying in the interior and boundary of the existence region
respectively.
  We also address the effect of requiring the partition to be a deterministic
function (i.e. a factor) of the points. Here we find the optimal tail behaviour
in dimension 1. There is a further separation into two regimes, governed by
algebraic properties of the allowed family types.
