  We show that the Gaussian primes $P[i] \subseteq \Z[i]$ contain infinitely
constellations of any prescribed shape and orientation. More precisely, given
any distinct Gaussian integers $v_0,...,v_{k-1}$, we show that there are
infinitely many sets $\{a+rv_0,...,a+rv_{k-1}\}$, with $a \in \Z[i]$ and $r \in
\Z \backslash \{0\}$, all of whose elements are Gaussian primes.
  The proof is modeled on a recent paper by Green and Tao and requires three
ingredients. The first is a hypergraph removal lemma of Gowers and
R\"odl-Skokan; this hypergraph removal lemma can be thought of as a
generalization of the Szemer\'edi-Furstenberg-Katznelson theorem concerning
multidimensional arithmetic progressions. The second ingredient is the
transference argument of Green and Tao, which allows one to extend this
hypergraph removal lemma to a relative version, weighted by a pseudorandom
measure. The third ingredient is a Goldston-Yildirim type analysis for the
Gaussian integers, which yields a pseudorandom measure which is concentrated on
Gaussian "almost primes".
