We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to Σ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma ^0_2$$\end{document}-induction over RCA0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {RCA}_0$$\end{document}. The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees TT1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {T}\,}{\mathsf {T}}^1$$\end{document} with an extra condition on the solution tree. (shrink)

We present a list of open questions in reverse mathematics, including some relevant background information for each question. We also mention some of the areas of reverse mathematics that are starting to be developed and where interesting open question may be found.

We formulate a polarized version of Ramsey’s theorem for trees. For those exponents greater than 2, both the reverse mathematics and the computability theory associated with this theorem parallel that of its linear analog. For pairs, the situation is more complex. In particular, there are many reasonable notions of stability in the tree setting, complicating the analysis of the related results.

A partial ordering $\mathbb{P}$ is $n$-Ramsey if, for every coloring of $n$-element chains from $\mathbb{P}$ in finitely many colors, $\mathbb{P}$ has a homogeneous subordering isomorphic to $\mathbb{P}$. In their paper on Ramsey properties of the complete binary tree, Chubb, Hirst, and McNicholl ask about Ramsey properties of other partial orderings. They also ask whether there is some Ramsey property for pairs equivalent to $\mathit{ACA}_{0}$ over $\mathit{RCA}_{0}$. A characterization theorem for finite-level partial orderings with Ramsey properties has been proven by the (...) second author. We show, over $\mathit{RCA}_{0}$, that one direction of the equivalence given by this theorem is equivalent to $\mathit{ACA}_{0}$, and the other is provable in $\mathit{ATR}_{0}$. We answer Chubb, Hirst, and McNicholl’s second question by showing that there is a primitive recursive partial ordering $\mathbb{P}$ such that, over $\mathit{RCA}_{0}$, “$\mathbb{P}$ is 2-Ramsey” is equivalent to $\mathit{ACA}_{0}$. (shrink)

We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to \-induction over \. The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees \ with an extra condition on the solution tree.

We show that a version of Ramsey’s theorem for trees for arbitrary exponents is equivalent to the subsystem ${{\sf ACA}^\prime_{0}}$ of reverse mathematics.

We show that a version of Ramsey’s theorem for trees for arbitrary exponents is equivalent to the subsystem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\sf ACA}^\prime_{0}}$$\end{document} of reverse mathematics.