scholarly journals RAMSEY’S THEOREM FOR PAIRS ANDKCOLORS AS A SUB-CLASSICAL PRINCIPLE OF ARITHMETIC

2017 ◽  
Vol 82 (2) ◽  
pp. 737-753
Author(s):  
STEFANO BERARDI ◽  
SILVIA STEILA
Keyword(s):  
Natural Number ◽  
Ramsey's Theorem ◽  
Recursively Enumerable ◽  
Ramsey’S Theorem ◽  
Heyting Arithmetic ◽  
Classical Principle ◽  
Image Position ◽  
Intuitionistic Arithmetic

AbstractThe purpose is to study the strength of Ramsey’s Theorem for pairs restricted to recursive assignments ofk-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number$k \ge 2$, Ramsey’s Theorem for pairs and recursive assignments ofkcolors is equivalent to the Limited Lesser Principle of Omniscience for${\rm{\Sigma }}_3^0$formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinitek-ary tree there is some$i < k$and some branch with infinitely many children of indexi.

Register machine proof of the theorem on exponential diophantine representation of enumerable sets

1984 ◽  
Vol 49 (3) ◽  
pp. 818-829 ◽  
Author(s):  
J. P. Jones ◽  
Y. V. Matijasevič
Keyword(s):  
Natural Number ◽  
Polynomial Time ◽  
Simple Proof ◽  
Diophantine Equations ◽  
Polynomial Equation ◽  
Exponential Diophantine Equations ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Exponential Relation

The purpose of the present paper is to give a new, simple proof of the theorem of M. Davis, H. Putnam and J. Robinson [1961], which states that every recursively enumerable relation A(a1, …, an) is exponential diophantine, i.e. can be represented in the formwhere a1 …, an, x1, …, xm range over natural numbers and R and S are functions built up from these variables and natural number constants by the operations of addition, A + B, multiplication, AB, and exponentiation, AB. We refer to the variables a1,…,an as parameters and the variables x1 …, xm as unknowns.Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilbert's tenth problem by the second author [1970], who proved that the exponential relation, a = bc, is diophantine, and hence that the right side of (1) can be replaced by a polynomial equation. But this part will not be reproved here. Readers wishing to read about the proof of that are directed to the papers of Y. Matijasevič [1971a], M. Davis [1973], Y. Matijasevič and J. Robinson [1975] or C. Smoryński [1972]. We concern ourselves here for the most part only with exponential diophantine equations until §5 where we mention a few consequences for the class NP of sets computable in nondeterministic polynomial time.

An ordinal partition avoiding pentagrams

2000 ◽  
Vol 65 (3) ◽  
pp. 969-978 ◽  
Author(s):  
Jean A. Larson
Keyword(s):  
Independent Set ◽  
Order Type ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Image Position

AbstractSuppose that α = γ + δ where γ ≥ δ > 0. Then there is a graph which has no independent set of order type and has no pentagram (a pentagram is a set of five points with all pairs joined by edges). In the notation of Erdős and Rado. who generalized Ramsey's Theorem to this setting.

scholarly journals THE DEFINABILITY STRENGTH OF COMBINATORIAL PRINCIPLES

2016 ◽  
Vol 81 (4) ◽  
pp. 1531-1554 ◽  
Author(s):  
WEI WANG
Keyword(s):  
Combinatorial Problem ◽  
Reverse Mathematics ◽  
Ramsey's Theorem ◽  
Combinatorial Principle ◽  
Definable Set ◽  
Ramsey’S Theorem ◽  
Combinatorial Principles ◽  
Definition Of ◽  
Image Position

AbstractWe introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a definable set. We prove that some consequences of Ramsey’s Theorem for colorings of pairs could help in simplifying the definitions of some ${\rm{\Delta }}_2^0$ sets, while some others could not. We also investigate some consequences of Ramsey’s Theorem for colorings of longer tuples. These results of definability strength have some interesting consequences in reverse mathematics, including strengthening of known theorems in a more uniform way and also new theorems.

STRONG REDUCTIONS BETWEEN COMBINATORIAL PRINCIPLES

2016 ◽  
Vol 81 (4) ◽  
pp. 1405-1431 ◽  
Author(s):  
DAMIR D. DZHAFAROV
Keyword(s):  
Reverse Mathematics ◽  
Second Order ◽  
New Techniques ◽  
Ramsey's Theorem ◽  
Second Order Arithmetic ◽  
Ramsey’S Theorem ◽  
Problems And Solutions ◽  
Combinatorial Principles ◽  
Order Arithmetic ◽  
Image Position

AbstractThis paper is a contribution to the growing investigation of strong reducibilities between ${\rm{\Pi }}_2^1$ statements of second-order arithmetic, viewed as an extension of the traditional analysis of reverse mathematics. We answer several questions of Hirschfeldt and Jockusch [13] about Weihrauch (uniform) and strong computable reductions between various combinatorial principles related to Ramsey’s theorem for pairs. Among other results, we establish that the principle $SRT_2^2$ is not Weihrauch or strongly computably reducible to $D_{ < \infty }^2$, and that COH is not Weihrauch reducible to $SRT_{ < \infty }^2$, or strongly computably reducible to $SRT_2^2$. The last result also extends a prior result of Dzhafarov [9]. We introduce a number of new techniques for controlling the combinatorial and computability-theoretic properties of the problems and solutions we construct in our arguments.

Borel sets and Ramsey's theorem

1973 ◽  
Vol 38 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Fred Galvin ◽  
Karel Prikry
Keyword(s):  
Recursion Theory ◽  
Main Result Theorem ◽  
Borel Sets ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Power Set ◽  
Definable Sets ◽  
Result Theorem ◽  
Image Position ◽  
Usual Product

Definition 1. For a set S and a cardinal κ,In particular, 2ω denotes the power set of the natural numbers and not the cardinal 2ℵ0. We regard 2ω as a topological space with the usual product topology.Definition 2. A set S ⊆ 2ω is Ramsey if there is an M ∈ [ω]ω such that either [M]ω ⊆ S or else [M]ω ⊆ 2ω − S.Erdös and Rado [3, Example 1, p. 434] showed that not every S ⊆ 2ω is Ramsey. In view of the nonconstructive character of the counterexample, one might expect (as Dana Scott has suggested) that all sufficiently definable sets are Ramsey. In fact, our main result (Theorem 2) is that all Borei sets are Ramsey. Soare [10] has applied this result to some problems in recursion theory.The first positive result on Scott's problem was Ramsey's theorem [8, Theorem A]. The next advance was Nash-Williams' generalization of Ramsey's theorem (Corollary 2), which can be interpreted as saying: If S1 and S2 are disjoint open subsets of 2ω, there is an M ∈ [ω]ω such that either [M]ω ⋂ S1 = ∅ or [M]ω ∩ S2 = ⊆. (This is halfway between “clopen sets are Ramsey” and “open sets are Ramsey.”) Then Galvin [4] stated a generalization of Nash-Williams' theorem (Corollary 1) which says, in effect, that open sets are Ramsey; this was discovered independently by Andrzej Ehrenfeucht, Paul Cohen, and probably many others, but no proof has been published.

THE STRENGTH OF RAMSEY’S THEOREM FOR COLORING RELATIVELY LARGE SETS

2014 ◽  
Vol 79 (01) ◽  
pp. 89-102 ◽  
Author(s):  
LORENZO CARLUCCI ◽  
KONRAD ZDANOWSKI
Keyword(s):  
Current Knowledge ◽  
Type Theorem ◽  
Complete Characterization ◽  
Infinite Subset ◽  
Point Of View ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Arithmetical Truth ◽  
Image Position

Abstract We characterize the effective content and the proof-theoretic strength of a Ramsey-type theorem for bi-colorings of so-called exactly large sets. An exactly large set is a set $X \subset {\bf{N}}$ such that ${\rm{card}}\left( X \right) = {\rm{min}}\left( X \right) + 1$ . The theorem we analyze is as follows. For every infinite subset M of N, for every coloring C of the exactly large subsets of M in two colors, there exists and infinite subset L of M such that C is constant on all exactly large subsets of L. This theorem is essentially due to Pudlák and Rödl and independently to Farmaki. We prove that—over RCA0 —this theorem is equivalent to closure under the ωth Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. In terms of Reverse Mathematics we give the first Ramsey-theoretic characterization of ${\rm{ACA}}_0^ +$ . Our results give a complete characterization of the theorem from the point of view of Computability Theory and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey’s Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey’s Theorem. We conjecture that analogous results hold for larger ordinals.

scholarly journals ON THE UNIFORM COMPUTATIONAL CONTENT OF RAMSEY’S THEOREM

2017 ◽  
Vol 82 (4) ◽  
pp. 1278-1316 ◽  
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 ◽  
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.

scholarly journals THE STRENGTH OF THE TREE THEOREM FOR PAIRS IN REVERSE MATHEMATICS

2016 ◽  
Vol 81 (4) ◽  
pp. 1481-1499 ◽  
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.

The nonconstructive content of sentences of arithmetic

1978 ◽  
Vol 43 (3) ◽  
pp. 497-501
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Function Theory ◽  
First Order ◽  
The Standard Model ◽  
Image Position ◽  
The Relationship ◽  
Excluded Middle ◽  
Degrees Of Unsolvability ◽  
Intuitionistic Arithmetic

This note is concerned with the old topic, initiated by Kleene, of the connections between recursive function theory and provability in intuitionistic arithmetic. More specifically, we are interested in the relationship between the hierarchy of degrees of unsolvability and the interdeducibility of cases of excluded middle. The work described below was motivated by a counterexample, to be given presently, which shows that that relationship is more complicated than one might suppose.Let HA be first-order intuitionistic arithmetic. Let the symbol ⊢ mean derivability in HA. For each natural number n, let n¯ be the corresponding numeral. Let Ω be the standard model of arithmetic. Say that a sentence ϕ is true iff Ω⊨ ϕ. Now suppose ϕ(x) and Ψ(x) are formulas with only the variable x free. SupposeThen it is natural to conjecture that {n∣Ω⊨Ψ(n¯)} is recursive in {n∣Ω⊨ϕ(n¯)}.However, this conjecture is false. Consider the formula is a formalization of Kleene's T-predicate.

A theorem on shortening the length of proof in formal systems of arithmetic

1975 ◽  
Vol 40 (3) ◽  
pp. 398-400 ◽  
Author(s):  
Robert A. di Paola
Keyword(s):  
Ab Initio ◽  
Natural Number ◽  
A Priori ◽  
Relative Length ◽  
Constant Term ◽  
Recursive Functions ◽  
Formal Systems ◽  
Recursively Enumerable ◽  
Image Position ◽  
Standard Enumeration

This note is concerned with an aspect of the length of proof of formulas in recursively enumerable theories T adequate for recursive arithmetic. In particular, we consider the relative length of proof of formulas in the theories T and T(S), where F represents an r.e. set A in T and T(S) is the theory obtained from T by adjunction, as a new axiom, of a sentence S undecidable in T.Throughout the sequel T is a consistent, r.e. theory with standard formalization [7] in which all recursive functions of one variable are definable, and in which there is a binary formula x ≤ satisfying the well-known conditions [7]:Here is the constant term corresponding to the natural number n. Wn is the nth r.e. set in a standard enumeration of the r.e. sets. Also, we assume an a priori Gödel numbering of our formalism satisfying the usual conditions, so that all formulas are numbers ab initio.In the more common applications of the theorem below, if F is a k-ary formula of T, is a natural number that measures in some way the length of the shortest proof of in T.

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