  We define a weak notion of universality in symbolic dynamics and, by
generalizing a proof of Mike Hochman, we prove that this yields necessary
conditions on the forbidden patterns defining a universal subshift: These
forbidden patterns are necessarily in a Medvedev degree greater or equal than
the degree of the set of subshifts for which it is universal.
  We also show that this necessary condition is optimal by giving constructions
of universal subshifts in the same Medvedev degree as the subshifts they
simulate and prove that this universality can be achieved by the sofic
projective subdynamics of such a subshift as soon as the necessary conditions
are verified.
  This could be summarized as: There are obstructions for the existence of
universal subshifts due to the theory of computability and they are the only
ones.
