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

Ramsey-type graph coloring and diagonal non-computability

A function is diagonally non-computable (d.n.c.) if it diagonalizes against the universal partial computable function. D.n.c. functions play a central role in algorithmic randomness and reverse mathematics. Flood and Towsner asked for which functions h, the principle stating the existence of an h-bounded d.n.c. function (DNR_h) implies the Ramsey-type K\"onig's lemma (RWKL). In this paper, we prove that for every computable order h, there exists an~$\omega$-model of DNR_h which is not a not model of the Ramsey-type graph coloring principle for two colors (RCOLOR2) and therefore not a model of RWKL. The proof combines bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to transform a computable non-reducibility into a separation over omega-models.Comment: 18 page

The weakness of being cohesive, thin or free in reverse mathematics

Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey's theorem and its consequence under the frameworks of reverse mathematics and computable reducibility. To this end, we study the degrees of unsolvability of cohesive sets for different uniformly computable sequence of sets and identify different layers of unsolvability. This analysis enables us to answer some questions of Wang about how typical sets help computing cohesive sets. We also study the impact of the number of colors in the computable reducibility between coloring statements. In particular, we strengthen the proof by Dzhafarov that cohesiveness does not strongly reduce to stable Ramsey's theorem for pairs, revealing the combinatorial nature of this non-reducibility and prove that whenever $k$ is greater than $\ell$, stable Ramsey's theorem for $n$-tuples and $k$ colors is not computably reducible to Ramsey's theorem for $n$-tuples and $\ell$ colors. In this sense, Ramsey's theorem is not robust with respect to his number of colors over computable reducibility. Finally, we separate the thin set and free set theorem from Ramsey's theorem for pairs and identify an infinite decreasing hierarchy of thin set theorems in reverse mathematics. This shows that in reverse mathematics, the strength of Ramsey's theorem is very sensitive to the number of colors in the output set. In particular, it enables us to answer several related questions asked by Cholak, Giusto, Hirst and Jockusch.Comment: 31 page

Dominating the Erdos-Moser theorem in reverse mathematics

The Erdos-Moser theorem (EM) states that every infinite tournament has an infinite transitive subtournament. This principle plays an important role in the understanding of the computational strength of Ramsey's theorem for pairs (RT^2_2) by providing an alternate proof of RT^2_2 in terms of EM and the ascending descending sequence principle (ADS). In this paper, we study the computational weakness of EM and construct a standard model (omega-model) of simultaneously EM, weak K\"onig's lemma and the cohesiveness principle, which is not a model of the atomic model theorem. This separation answers a question of Hirschfeldt, Shore and Slaman, and shows that the weakness of the Erdos-Moser theorem goes beyond the separation of EM from ADS proven by Lerman, Solomon and Towsner.Comment: 36 page

The weakness of the pigeonhole principle under hyperarithmetical reductions

The infinite pigeonhole principle for 2-partitions ($\mathsf{RT}^1_2$) asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that $\mathsf{RT}^1_2$ admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every $\Delta^0_n$ set, of an infinite low${}_n$ subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.Comment: 29 page

Pi01 encodability and omniscient reductions

A set of integers $A$ is computably encodable if every infinite set of integers has an infinite subset computing $A$. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this paper, we extend this notion of computable encodability to subsets of the Baire space and we characterize the $\Pi^0_1$ encodable compact sets as those who admit a non-empty $\Sigma^1_1$ subset. Thanks to this equivalence, we prove that weak weak K\"onig's lemma is not strongly computably reducible to Ramsey's theorem. This answers a question of Hirschfeldt and Jockusch.Comment: 9 page

Coloring trees in reverse mathematics

The tree theorem for pairs ($\mathsf{TT}^2_2$), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree $2^{<\omega}$, there is a set of nodes isomorphic to $2^{<\omega}$ which is homogeneous for the coloring. This is a generalization of the more familiar Ramsey's theorem for pairs ($\mathsf{RT}^2_2$), which has been studied extensively in computability theory and reverse mathematics. We answer a longstanding open question about the strength of $\mathsf{TT}^2_2$, by showing that this principle does not imply the arithmetic comprehension axiom ($\mathsf{ACA}_0$) over the base system, recursive comprehension axiom ($\mathsf{RCA}_0$), of second-order arithmetic. In addition, we give a new and self-contained proof of a recent result of Patey that $\mathsf{TT}^2_2$ is strictly stronger than $\mathsf{RT}^2_2$. Combined, these results establish $\mathsf{TT}^2_2$ as the first known example of a natural combinatorial principle to occupy the interval strictly between $\mathsf{ACA}_0$ and $\mathsf{RT}^2_2$. The proof of this fact uses an extension of the bushy tree forcing method, and develops new techniques for dealing with combinatorial statements formulated on trees, rather than on $\omega$.Comment: 25 page