Open questions about Ramsey-type statements in reverse mathematics

Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey's theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.Comment: 15 page

Controlling iterated jumps of solutions to combinatorial problems

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.Comment: 32 page

Iterative forcing and hyperimmunity in reverse mathematics

The separation between two theorems in reverse mathematics is usually done by constructing a Turing ideal satisfying a theorem P and avoiding the solutions to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a forcing technique for iterating a computable non-reducibility in order to separate theorems over omega-models. In this paper, we present a modularized version of their framework in terms of preservation of hyperimmunity and show that it is powerful enough to obtain the same separations results as Wang did with his notion of preservation of definitions.Comment: 15 page

Degrees bounding principles and universal instances in reverse mathematics

A Turing degree d bounds a principle P of reverse mathematics if every computable instance of P has a d-computable solution. P admits a universal instance if there exists a computable instance such that every solution bounds P. We prove that the stable version of the ascending descending sequence principle (SADS) as well as the stable version of the thin set theorem for pairs (STS(2)) do not admit a bound of low_2 degree. Therefore no principle between Ramsey's theorem for pairs RT22 and SADS or STS(2) admit a universal instance. We construct a low_2 degree bounding the Erd\H{o}s-Moser theorem (EM), thereby showing that previous argument does not hold for EM. Finally, we prove that the only Delta^0_2 degree bounding a stable version of the rainbow Ramsey theorem for pairs (SRRT22) is 0'. Hence no principle between the stable Ramsey theorem for pairs SRT22 and SRRT22 admit a universal instance. In particular the stable version of the Erd\H{o}s-Moser theorem does not admit one. It remains unknown whether EM admits a universal instance.Comment: 23 page