scholarly journals Generalized cohesiveness

1999 ◽  
Vol 64 (2) ◽  
pp. 489-516 ◽  
Author(s):  
Tamara Hummel ◽  
Carl G. Jockusch
Keyword(s):  
Natural Numbers ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Infinite Set ◽  
Degrees Of Unsolvability ◽  
Computably Enumerable

AbstractWe study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show that for all n ≥ 2, there exists a n-cohesive set. We improve this result for n = 2 by showing that there is a 2-cohesive set. We show that the n-cohesive and n-r-cohesive degrees together form a linear, non-collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0(n+1) and also characterize the jumps of the n-r-cohesive degrees.

Effective versions of Ramsey's Theorem: Avoiding the cone above 0′

1994 ◽  
Vol 59 (4) ◽  
pp. 1301-1325 ◽  
Author(s):  
Tamara Lakins Hummel
Keyword(s):  
Reverse Mathematics ◽  
Natural Numbers ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Homogeneous Set ◽  
Recent Theorem ◽  
Infinite Set ◽  
Degrees Of Unsolvability

AbstractRamsey's Theorem states that if P is a partition of [ω]k into finitely many partition classes, then there exists an infinite set of natural numbers which is homogeneous for P. We consider the degrees of unsolvability and arithmetical definability properties of infinite homogeneous sets for recursive partitions. We give Jockusch's proof of Seetapun's recent theorem that for all recursive partitions of [ω]2 into finitely many pieces, there exists an infinite homogeneous set A such that ∅′ ≰TA. Two technical extensions of this result are given, establishing arithmetical bounds for such a set A. Applications to reverse mathematics and introreducible sets are discussed.

Ramsey's theorem for computably enumerable colorings

2001 ◽  
Vol 66 (2) ◽  
pp. 873-880 ◽  
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.

Types omitted in uncountable models of arithmetic

1975 ◽  
Vol 40 (3) ◽  
pp. 317-320 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Completeness Theorem ◽  
Elementary Extension ◽  
Ramsey's Theorem ◽  
Definable Set ◽  
Ramsey’S Theorem ◽  
Hanf Number ◽  
Definable Sets ◽  
Omitting Types ◽  
Infinite Set ◽  
Models Of Arithmetic

In [4] it is shown that if the structure omits a type Σ, and Σ is complete with respect to Th(), then there is a proper elementary extension of which omits Σ. This result is extended in the present paper. It is shown that Th() has models omitting Σ in all infinite powers.A type is a countable set of formulas with just the variable ν occurring free. A structure is said to omit the type Σ if no element of satisfies all of the formulas of Σ. A type Σ, in the same language as a theory T, is said to be complete with respect to T if (1) T ∪ Σ is consistent, and (2) for every formula φ(ν) of the language of T (with just ν free), either φ or ¬φ is in Σ.The proof of the result of this paper resembles Morley's proof [5] that the Hanf number for omitting types is . It is shown that there is a model of Th() which omits Σ and contains an infinite set of indiscernibles. Where Morley used the Erdös-Rado generalization of Ramsey's theorem, a definable version of the ordinary Ramsey's theorem is used here.The “omitting types” version of the ω-completeness theorem ([1], [3], [6]) is used, as it was in Morley's proof and in [4]. In [4], satisfaction of the hypotheses of the ω-completeness theorem followed from the fact that, in , any infinite, definable set can be split into two infinite, definable sets.

Sets which do not have subsets of every higher degree

1978 ◽  
Vol 43 (1) ◽  
pp. 135-138 ◽  
Author(s):  
Stephen G. Simpson
Keyword(s):  
Similar Degree ◽  
Theoretic Analysis ◽  
Natural Numbers ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Infinite Sets ◽  
Special Case

Let A be a subset of ω, the set of natural numbers. The degree of A is its degree of recursive unsolvability. We say that A is rich if every degree above that of A is represented by a subset of A. We say that A is poor if no degree strictly above that of A is represented by a subset of A. The existence of infinite poor (and hence nonrich) sets was proved by Soare [9].Theorem 1. Suppose that A is infinite and not rich. Then every hyperarith-metical subset H of ω is recursive in A.In the special case when H is arithmetical, Theorem 1 was proved by Jockusch [4] who employed a degree-theoretic analysis of Ramsey's theorem [3]. In our proof of Theorem 1 we employ a similar, degree-theoretic analysis of a certain generalization of Ramsey's theorem. The generalization of Ramsey's theorem is due to Nash-Williams [6]. If A ⊆ ω we write [A]ω for the set of all infinite subsets of A. If P ⊆ [ω]ω we let H(P) be the set of all infinite sets A such that either [A]ω ⊆ P = ∅. Nash-Williams' theorem is essentially the statement that if P ⊆ [ω]ω is clopen (in the usual, Baire topology on [ω]ω) then H(P) is nonempty. Subsequent, further generalizations of Ramsey's theorem were proved by Galvin and Prikry [1], Silver [8], Mathias [5], and analyzed degree-theoretically by Solovay [10]; those results are not needed for this paper.

Ramsey's theorem and recursion theory

1972 ◽  
Vol 37 (2) ◽  
pp. 268-280 ◽  
Author(s):  
Carl G. Jockusch
Keyword(s):  
Recursion Theory ◽  
The Other ◽  
Natural Numbers ◽  
Arithmetical Hierarchy ◽  
Original Proof ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Infinite Sets ◽  
Recursive Partition ◽  
Positive Results

Let N be the set of natural numbers. If A ⊆ N, let [A]n denote the class of all n-element subsets of A. If P is a partition of [N]n into finitely many classes C1, …, Cp, let H(P) denote the class of those infinite sets A ⊆ N such that [A]n ⊆ Ci for some i. Ramsey's theorem [8, Theorem A] asserts that H(P) is nonempty for any such partition P. Our purpose here is to study what can be said about H(P) when P is recursive, i.e. each Ci, is recursive under a suitable coding of [N]n. We show that if P is such a recursive partition of [N]n, then H(P) contains a set which is Πn0 in the arithmetical hierarchy. In the other direction we prove that for each n ≥ 2 there is a recursive partition P of [N]n into two classes such that H(P) contains no Σn0 set. These results answer a question raised by Specker [12].A basic partition is a partition of [N]2 into two classes. In §§2, 3, and 4 we concentrate on basic partitions and in so doing prepare the way for the general results mentioned above. These are proved in §5. Our “positive” results are obtained by effectivizing proofs of Ramsey's theorem which differ from the original proof in [8]. We present these proofs (of which one is a generalization of the other) in §§4 and 5 in order to clarify the motivation of the effective versions.

Ramsey's theorem in the hierarchy of choice principles

1977 ◽  
Vol 42 (3) ◽  
pp. 387-390 ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Axiom Of Choice ◽  
Extension Principle ◽  
Infinite Length ◽  
Finite Sets ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
The Axiom Of Choice ◽  
Infinite Set ◽  
Dependent Choice ◽  
Weak Axioms Of Choice

Ramsey's theorem [5] asserts that every infinite set X has the following partition property (RP): For every partition of the set [X]2 of two-element subsets of X into two pieces, there is an infinite subset Y of X such that [Y]2 is included in one of the pieces. Ramsey explicitly indicated that his proof of this theorem used the axiom of choice. Kleinberg [3] showed that every proof of Ramsey's theorem must use the axiom of choice, although rather weak forms of this axiom suffice. J. Dawson has raised the question of the position of Ramsey's theorem in the hierarchy of weak axioms of choice.In this paper, we prove or refute the provability of each of the possible implications between Ramsey's theorem and the weak axioms of choice mentioned in Appendix A.3 of Jech's book [2]. Our results, along with some known facts which we include for completeness, may be summarized as follows (the notation being as in [2]):A. The following principles do not (even jointly) imply Ramsey's theorem, nor does Ramsey's theorem imply any of them:the Boolean prime ideal theorem,the selection principle,the order extension principle,the ordering principle,choice from wellordered sets (ACW),choice from finite sets,choice from pairs (C2).B. Each of the following principles implies Ramsey's theorem, but none of them follows from Ramsey's theorem:the axiom of choice,wellordered choice (∀kACk),dependent choice of any infinite length k (DCk),countable choice (ACN0),nonexistence of infinite Dedekind-finite sets (WN0).

scholarly journals OPEN QUESTIONS ABOUT RAMSEY-TYPE STATEMENTS IN REVERSE MATHEMATICS

2016 ◽  
Vol 22 (2) ◽  
pp. 151-169 ◽  
Author(s):  
LUDOVIC PATEY
Keyword(s):  
Reverse Mathematics ◽  
Ramsey's Theorem ◽  
The Past ◽  
Open Questions ◽  
Ramsey’S Theorem ◽  
Ramsey Type ◽  
Infinite Set

AbstractRamsey’s theorem states that for any coloring of then-element subsets of ℕ with finitely many colors, there is an infinite setHsuch that alln-element subsets ofHhave 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.

scholarly journals On the strength of Ramsey's theorem for pairs

2001 ◽  
Vol 66 (1) ◽  
pp. 1-55 ◽  
Author(s):  
Peter A. Cholak ◽  
Carl G. Jockusch ◽  
Theodore A. Slaman
Keyword(s):  
Natural Numbers ◽  
Ramsey's Theorem ◽  
Strengthened Version ◽  
Ramsey’S Theorem ◽  
Homogeneous Set ◽  
Models Of Arithmetic

AbstractWe study the proof–theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RTkn denote Ramsey's theorem for k–colorings of n–element sets, and let RT<∞n denote (∀k)RTkn. Our main result on computability is: For any n ≥ 2 and any computable (recursive) k–coloring of the n–element sets of natural numbers, there is an infinite homogeneous set X with X″ ≤T 0(n). Let IΣn and BΣn denote the Σn induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low2) to models of arithmetic enables us to show that RCA0 + IΣ2 + RT22 is conservative over RCA0 + IΣ2 for Π11 statements and that RCA0 + IΣ3 + RT<∞2 is Π11-conservative over RCA0 + IΣ3. It follows that RCA0 + RT22 does not imply BΣ3. In contrast, J. Hirst showed that RCA0 + RT<∞2 does imply BΣ3, and we include a proof of a slightly strengthened version of this result. It follows that RT<∞2 is strictly stronger than RT22 over RCA0.

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

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
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