The strength of Ramsey’s theorem for pairs over trees: I. Weak König’s lemma

10.2178/jsl.7704120 ◽

2012 ◽

Vol 77 (4) ◽

pp. 1272-1280 ◽

Cited By ~ 9

Author(s):

Stephen Flood

Keyword(s):

Reverse Mathematics ◽

Ramsey's Theorem ◽

Weak Regularity ◽

Ramsey’S Theorem ◽

Ramsey Type ◽

Weak König’S Lemma ◽

AbstractIn this paper, we propose a weak regularity principle which is similar to both weak König's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then analyze different ways of generalizing this principle.

ON THE UNIFORM COMPUTATIONAL CONTENT OF RAMSEY’S THEOREM

10.1017/jsl.2017.43 ◽

2017 ◽

Vol 82 (4) ◽

pp. 1278-1316 ◽

Cited By ~ 6

Author(s):

VASCO BRATTKA ◽

TAHINA RAKOTONIAINA

Keyword(s):

Finite Number ◽

Ramsey's Theorem ◽

Borel Hierarchy ◽

Ramsey’S Theorem ◽

Homogeneous Set ◽

Colored Version ◽

Output Information ◽

Image Position ◽

Weak König’S Lemma ◽

AbstractWe study the uniform computational content of Ramsey’s theorem in the Weihrauch lattice. Our central results provide information on how Ramsey’s theorem behaves under product, parallelization, and jumps. From these results we can derive a number of important properties of Ramsey’s theorem. For one, the parallelization of Ramsey’s theorem for cardinalityn≥ 1 and an arbitrary finite number of colorsk≥ 2 is equivalent to then-th jump of weak Kőnig’s lemma. In particular, Ramsey’s theorem for cardinalityn≥ 1 is${\bf{\Sigma }}_{n + 2}^0$-measurable in the effective Borel hierarchy, but not${\bf{\Sigma }}_{n + 1}^0$-measurable. Secondly, we obtain interesting lower bounds, for instance then-th jump of weak Kőnig’s lemma is Weihrauch reducible to (the stable version of) Ramsey’s theorem of cardinalityn+ 2 forn≥ 2. We prove that with strictly increasing numbers of colors Ramsey’s theorem forms a strictly increasing chain in the Weihrauch lattice. Our study of jumps also shows that certain uniform variants of Ramsey’s theorem that are indistinguishable from a nonuniform perspective play an important role. For instance, the colored version of Ramsey’s theorem explicitly includes the color of the homogeneous set as output information, and the jump of this problem (but not the uncolored variant) is equivalent to the stable version of Ramsey’s theorem of the next greater cardinality. Finally, we briefly discuss the particular case of Ramsey’s theorem for pairs, and we provide some new separation techniques for problems that involve jumps in this context. In particular, we study uniform results regarding the relation of boundedness and induction problems to Ramsey’s theorem, and we show that there are some significant differences with the nonuniform situation in reverse mathematics.

THE STRENGTH OF THE TREE THEOREM FOR PAIRS IN REVERSE MATHEMATICS

10.1017/jsl.2015.80 ◽

2016 ◽

Vol 81 (4) ◽

pp. 1481-1499 ◽

Cited By ~ 1

Author(s):

LUDOVIC PATEY

Keyword(s):

Binary Tree ◽

Reverse Mathematics ◽

Theoretic Property ◽

Ramsey's Theorem ◽

Ramsey’S Theorem ◽

Image Position ◽

Weak König’S Lemma ◽

König’S Lemma ◽

Comprehension Axiom

AbstractNo natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA0) 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 ACA0 and $RT_2^2$ over RCA0, but its exact strength remains open. In this paper, we prove that $RT_2^2$ together with weak König’s lemma (WKL0) does not imply $TT_2^2$, thereby answering a question of Montálban. This separation is a case in point of the method of Lerman, Solomon and Towsner for designing a computability-theoretic property which discriminates between two statements in reverse mathematics. We therefore put the emphasis on the different steps leading to this separation in order to serve as a tutorial for separating principles in reverse mathematics.

Ramsey’s theorem and König’s Lemma

Archive for Mathematical Logic ◽

10.1007/s00153-006-0025-z ◽

2006 ◽

Vol 46 (1) ◽

pp. 37-42 ◽

Cited By ~ 3

Author(s):

T. E. Forster ◽

J. K. Truss

Keyword(s):

Ramsey's Theorem ◽

Ramsey’S Theorem ◽

On notions of computability-theoretic reduction between Π21 principles

Journal of Mathematical Logic ◽

10.1142/s0219061316500021 ◽

2016 ◽

Vol 16 (01) ◽

pp. 1650002 ◽

Cited By ~ 7

Author(s):

Denis R. Hirschfeldt ◽

Carl G. Jockusch

Keyword(s):

Ramsey's Theorem ◽

Ramsey’S Theorem ◽

Homogeneous Set ◽

Thin Set ◽

Several notions of computability-theoretic reducibility between [Formula: see text] principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each [Formula: see text], there is an instance of RT[Formula: see text] all of whose solutions have PA degree over [Formula: see text] and use this to show that König’s Lemma lies strictly between RT[Formula: see text] and RT[Formula: see text] under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer (2016) on comparing versions of Ramsey’s Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey-theoretic properties, we show that for each [Formula: see text], there is an [Formula: see text]-coloring [Formula: see text] of [Formula: see text] such that every [Formula: see text]-coloring of [Formula: see text] has an infinite homogeneous set that does not compute any infinite homogeneous set for [Formula: see text], and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa (to appear). Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to [Formula: see text]-models of RCA0, and related notions that allow us to count how many applications of a principle [Formula: see text] are needed to reduce another principle to [Formula: see text]. Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman (2001).

The Brouwer invariance theorems in reverse mathematics

Forum of Mathematics Sigma ◽

10.1017/fms.2020.52 ◽

2020 ◽

Vol 8 ◽

Author(s):

Takayuki Kihara

Keyword(s):

Reverse Mathematics ◽

Base System ◽

Open Problems ◽

Explicit Algorithm ◽

Topological Embedding ◽

Weak König’S Lemma ◽

Abstract In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ In this article, we solve Stillwell’s problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König’s lemma over the base system ${\sf RCA}_0$ . In particular, there exists an explicit algorithm which, whenever the weak König’s lemma is false, constructs a topological embedding of $\mathbb {R}^4$ into $\mathbb {R}^3$ .

Ramsey’s Theorem

The Discrete Mathematical Charms of Paul Erdős ◽

10.1017/9781108912181.006 ◽

2021 ◽

pp. 36-50

Keyword(s):

Ramsey's Theorem ◽

Ramsey’S Theorem

Ramsey's theorem for computably enumerable colorings

10.2307/2695050 ◽

2001 ◽

Vol 66 (2) ◽

pp. 873-880 ◽

Cited By ~ 1

Author(s):

Tamara J. Hummel ◽

Carl G. Jockusch

Keyword(s):

Analogous Result ◽

Computably Enumerable Set ◽

Ramsey's Theorem ◽

Ramsey’S Theorem ◽

Computably Enumerable

AbstractIt is shown that for each computably enumerable set of n-element subsets of ω there is an infinite set A ⊆ ω such that either all n-element subsets of A are in or no n-element subsets of A are in . An analogous result is obtained with the requirement that A be replaced by the requirement that the jump of A be computable from 0(n). These results are best possible in various senses.

Weaker cousins of Ramsey's theorem over a weak base theory

Annals of Pure and Applied Logic ◽

10.1016/j.apal.2021.103028 ◽

2021 ◽

pp. 103028

Author(s):

Marta Fiori-Carones ◽

Leszek Aleksander Kołodziejczyk ◽

Katarzyna W. Kowalik

Keyword(s):

Weak Base ◽

Ramsey's Theorem ◽

Base Theory ◽

Ramsey’S Theorem

Combinatorial principles weaker than Ramsey's Theorem for pairs

10.2178/jsl/1174668391 ◽

2007 ◽

Vol 72 (1) ◽

pp. 171-206 ◽

Cited By ~ 47

Author(s):

Denis R. Hirschfeldt ◽

Richard A. Shore

Keyword(s):

Reverse Mathematics ◽

The Other ◽

Base System ◽

Ramsey's Theorem ◽

Open Questions ◽

Ramsey’S Theorem ◽

Combinatorial Principles ◽

Points Of View ◽

The One ◽

Infinite Linear

AbstractWe investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs () splits into a stable version () and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for and ). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2. We show, for instance, that WKL0 is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is -conservative over the base system RCA0. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch. Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply (and so does not imply ). This answers a question of Cholak, Jockusch, and Slaman.Our proofs suggest that the essential distinction between ADS and CAC on the one hand and on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions.