A packed Ramsey's theorem and computability theory

Partial orders and immunity in reverse mathematics

We identify computability-theoretic properties enabling us to separate various statements about partial orders in reverse mathematics. We obtain simpler proofs of existing separations, and deduce new compound ones. This work is part of a larger program of unification of the separation proofs of various Ramsey-type theorems in reverse mathematics in order to obtain a better understanding of the combinatorics of Ramsey's theorem and its consequences. We also answer a question of Murakami, Yamazaki and Yokoyama about pseudo Ramsey's theorem for pairs.

SRT22 does not imply RT22 in omega-models

We complete a 40-year old program on the computability-theoretic analysis of Ramsey's theorem, starting with Jockusch in 1972, and improving a result of Chong, Slaman and Yang in 2014. Given a set $X$, let $[X]^n$ be the collection of all $n$-element subsets of $X$. Ramsey's theorem for $n$-tuples asserts the existence, for every finite coloring of $[ω]^n$, of an infinite set $X \subseteq ω$ such that $[X]^n$ is monochromatic. The meta-mathematical study of Ramsey has a rich history, with several long-standing open problems and seminal theorems, including Seetapun's theorem in 1995 and Liu's t…

The Strength of Some Combinatorial Principles Related to Ramsey's Theorem for Pairs

We study the reverse mathematics and computability-the\-o\-re\-tic strength of (stable) Ramsey's Theorem for pairs and the related principles COH and DNR. We show that SRT$^2_2$ implies DNR over RCA$_0$ but COH does not, and answer a question of Mileti by showing that every computable stable $2$-coloring of pairs has an incomplete $Δ^0_2$ infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT$^2_2$ does not imply RT$^2_2$.

关于Ramsey定理对对的低基理定理

Low-like basis theorems for Ramsey's theorem for pairs in first-order arithmetic

本文在BΣ^0_3内构建了RT^2的\ll^2-解，并证明了其基理定理，同时展示了RT^2_2和EM_{<\infty}的类似定理，为相关保守性结果提供了更简单的证明。

Coarse computability, the density metric, Hausdorff distances between Turing degrees, perfect trees, and reverse mathematics

The coarse similarity class $[A]$ of $A$ is the set of all $B$ whose symmetric difference with $A$ has asymptotic density 0. There is a natural metric $δ$ on the space $\mathcal{S}$ of coarse similarity classes defined by letting $δ([A],[B])$ be the upper density of the symmetric difference of $A$ and $B$. We study the resulting metric space, showing in particular that between any two distinct points there are continuum many geodesic paths. We also study subspaces of the form $\{[A] : A \in \mathcal U\}$ where $\mathcal U$ is closed under Turing equivalence, and show that there is a tight conn…

Ginsburg--Sands 定理与可计算性理论

The Ginsburg--Sands theorem and computability theory

本文在可计算性理论和逆向数学中分析了拓扑学的Ginsburg-Sands定理，证明了其在CSC、Hausdorff和T1等不同空间限制下分别等价于ACA0、CAC等系统，并发现T1情形严格介于ACA0与RT2_2之间，同时引入了新的组合原理层级。

A computable analysis of variable words theorems

The Carlson-Simpson lemma is a combinatorial statement occurring in the proof of the Dual Ramsey theorem. Formulated in terms of variable words, it informally asserts that given any finite coloring of the strings, there is an infinite sequence with infinitely many variables such that for every valuation, some specific set of initial segments is homogeneous. Friedman, Simpson, and Montalban asked about its reverse mathematical strength. We study the computability-theoretic properties and the reverse mathematics of this statement, and relate it to the finite union theorem. In particular, we prov…

鸽巢分层的反演数学

The reverse mathematics of the pigeonhole hierarchy

本文从可计算性理论与反演数学视角，证明算术分层层级诱导的鸽巢原理分层在$\text{RCA}_0$上严格，研究其一阶后承，属组合定理计算内容的元数学研究。

The strength of the tree theorem for pairs in reverse mathematics

No natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA) and Ramsey's theorem for pairs (RT^2_2) in reverse mathematics. The tree theorem for pairs (TT^2_2) is however a good candidate. The tree theorem states that for every finite coloring over tuples of comparable nodes in the full binary tree, there is a monochromatic subtree isomorphic to the full tree. The principle TT^2_2 is known to lie between ACA and RT^2_2 over RCA, but its exact strength remains open. In this paper, we prove that RT^2_2 together with weak König's lemma (WKL) does not im…

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 $2^{<ω}$, 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 ($\mathsf{RT}^2_2$), 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 arithmeti…

Cohesive avoidance and arithmetical sets

An open question in reverse mathematics is whether the cohesive principle, $\COH$, is implied by the stable form of Ramsey's theorem for pairs, $\SRT^2_2$, in $ω$-models of $\RCA$. One typical way of establishing this implication would be to show that for every sequence $\vec{R}$ of subsets of $ω$, there is a set $A$ that is $Δ^0_2$ in $\vec{R}$ such that every infinite subset of $A$ or $\bar{A}$ computes an $\vec{R}$-cohesive set. In this article, this is shown to be false, even under far less stringent assumptions: for all natural numbers $n \geq 2$ and $m < 2^n$, there is a sequence $\vec{R…

The reverse mathematics of Hindman's theorem for sums of exactly two elements

Hindman's Theorem (HT) states that for every coloring of $\mathbb N$ with finitely many colors, there is an infinite set $H \subseteq \mathbb N$ such that all nonempty sums of distinct elements of $H$ have the same color. The investigation of restricted versions of HT from the computability-theoretic and reverse-mathematical perspectives has been a productive line of research recently. In particular, HT$^{\leqslant n}_k$ is the restriction of HT to sums of at most $n$ many elements, with at most $k$ colors allowed, and HT$^{=n}_k$ is the restriction of HT to sums of \emph{exactly} $n$ many ele…