Let A be a subset of positive relative upper density of \PP^d, the
d-tuples of primes. We prove that A contains an affine copy of any finite
set F\subs\Z^d, which provides a natural multi-dimensional extension of the
theorem of Green and Tao on the existence of long arithmetic progressions in
the primes. The proof uses the hypergraph approach by assigning a pseudo-random
weight system to the pattern F on a d+1-partite hypergraph; a novel feature
being that the hypergraph is no longer uniform with weights attached to lower
dimensional edges. Then, instead of using a transference principle, we proceed
by extending the proof of the so-called hypergraph removal lemma to our
settings, relying only on the linear forms condition of Green and Tao