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

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

Comparing the strength of diagonally non-recursive functions in the absence of Sigma^0_2 induction

We prove that the statement “there is a k such that for every f there is a k-bounded diagonally nonrecursive function relative to f” does not imply weak König’s lemma over . This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every k-bounded diagonally nonrecursive function computes a 2-bounded diagonally nonrecursive function may fail in the absence of

Coloring trees in reverse mathematics

International audienceThe tree theorem for pairs (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 <ω , there is a set of nodes isomorphic to 2 <ω which is homogeneous for the coloring. This is a generalization of the more familiar Ramsey's theorem for pairs (RT 2 2), which has been studied extensively in computability theory and reverse mathematics. We answer a longstanding open question about the strength of TT 2 2 , by showing that this principle does not imply the arithmetic comprehension axiom (ACA 0) over the base system, recursive comprehension axiom (RCA 0), of second-order arithmetic. In addition , we give a new and self-contained proof of a recent result of Patey that TT 2 2 is strictly stronger than RT 2 2. Combined, these results establish TT 2 2 as the first known example of a natural combinatorial principle to occupy the interval strictly between ACA 0 and 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 ω