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.

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.

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.

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.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

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

Combinatorial principles weaker than Ramsey's Theorem for pairs

2007 ◽  
Vol 72 (1) ◽  
pp. 171-206 ◽  
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.

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