Free sets, thin sets and rainbows for barriers

We formulate and prove the generalizations of Friedman’s free set and thin set theorems and of the rainbow Ramsey theorem to colorings of barriers. We analyze the strength of these theorems from the point of view of computability theory proving some upper and lower bounds on the complexity of solutions for computable instances and some uniform computable reductions. We obtain as corollaries some proof-theoretical results on the logical strength of the theorems, in the spirit of reverse mathematics.