  Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set,
the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to
collapse in reverse mathematics. A promising approach to show the strictness of
the hierarchies would be to prove that every computable instance at level n has
a low_n solution. In particular, this requires effective control of iterations
of the Turing jump. In this paper, we design some variants of Mathias forcing
to construct solutions to cohesiveness, the Erdos-Moser theorem and stable
Ramsey's theorem for pairs, while controlling their iterated jumps. For this,
we define forcing relations which, unlike Mathias forcing, have the same
definitional complexity as the formulas they force. This analysis enables us to
answer two questions of Wei Wang, namely, whether cohesiveness and the
Erdos-Moser theorem admit preservation of the arithmetic hierarchy, and can be
seen as a step towards the resolution of the strictness of the Ramsey-type
hierarchies.
