Controlling iterated jumps of solutions to combinatorial problems

TL;DR

This paper develops variants of Mathias forcing to construct solutions to combinatorial problems with controlled iterated Turing jumps, advancing understanding of the hierarchies in reverse mathematics.

Contribution

It introduces new forcing techniques that manage the complexity of solutions' jumps, addressing open questions in the structure of Ramsey-type hierarchies.

Findings

Constructed solutions with controlled iterated jumps for cohesiveness and Erdos-Moser theorem.

Answered questions about preservation of the arithmetic hierarchy for these theorems.

Provided tools for analyzing the strictness of Ramsey-type hierarchies.

Abstract

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.