相关论文: $\Pi^0_4$ conservation of Ramsey's theorem for pai…

In this article, we prove that Ramsey's theorem for pairs and two colors is $\Pi^1_1$-conservative over~$\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2 + \mathsf{WF}(\epsilon_0)$ and over~$\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2 + \bigcup_n…

Ramsey's theorem for $n$-tuples and $k$-colors ($\mathsf{RT}^n_k$) asserts that every k-coloring of $[\mathbb{N}]^n$ admits an infinite monochromatic subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and two…

In this paper, we show that $\mathrm{RT}^{2}+\mathsf{WKL}_0$ is a $\Pi^{1}_{1}$-conservative extension of $\mathrm{B}\Sigma^0_3$.

We study Ramsey's theorem for pairs and two colours in the context of the theory of $\alpha$-large sets introduced by Ketonen and Solovay. We prove that any $2$-colouring of pairs from an $\omega^{300n}$-large set admits an $\omega^n$-large…

Let $\mathsf{TT}^1$ be the combinatorial principle stating that every finite coloring of the infinite full binary tree has a homogeneous isomorphic subtree. Let $\mathsf{RT}^2_2$ and $\mathsf{WKL}_0$ denote respectively the principles of…

In this article, we study a degenerate version of Ramsey's theorem for pairs and two colors ($\mathsf{RT}^2_2$), in which the homogeneous sets for color 1 are of bounded size. By $\mathsf{RT}^2_2$, it follows that every such coloring admits…

We prove that any proof of a $\forall \Sigma^0_2$ sentence in the theory $\mathrm{WKL}_0 + \mathrm{RT}^2_2$ can be translated into a proof in $\mathrm{RCA}_0$ at the cost of a polynomial increase in size. In fact, the proof in…

Ramsey's theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey's theorem for pairs due to Erdos and Rado, which states that every…

The paper is devoted to a reverse-mathematical study of some well-known consequences of Ramsey's theorem for pairs, focused on the chain-antichain principle $\mathsf{CAC}$, the ascending-descending sequence principle $\mathsf{ADS}$, and the…

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…

We prove a sharp structural result concerning finite colorings of pairs in well-founded trees.

The purpose is to study the strength of Ramsey's Theorem for pairs restricted to recursive assignments of $k$-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number $k \geq 2$, Ramsey's…

The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are non-null in a…

We show that over the weak base theory $\mathrm{RCA}_0^*$, cohesive Ramsey's theorem for pairs $\mathrm{CRT}^2_2$ implies exponential closure of the definable cut $\mathrm{I}^0_1$, which is the intersection of all $\Sigma^0_1$-definable…

We study the strength of $\RRT^3_2$, Rainbow Ramsey Theorem for colorings of triples, and prove that $\RCA + \RRT^3_2$ implies neither $\WKL$ nor $\RRT^4_2$. To this end, we establish some recursion theoretic properties of cohesive sets and…

In \cite{J}, Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite $\Pi^0_2$ homogeneous set. The proof uses a countable collection of $\Pi^0_2$ sets as potential infinite homogeneous…

The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the…

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…

We prove that $\RCA + \RRT^3_2 \not\vdash \ACA$ where $\RRT^3_2$ is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs…

Let $\mathsf{TT}^2_k$ denote the combinatorial principle stating that every $k$-coloring of pairs of compatible nodes in the full binary tree has a homogeneous solution, i.e. an isomorphic subtree in which all pairs of compatible nodes have…