scholarly journals Π11-conservation of combinatorial principles weaker than Ramsey’s theorem for pairs

2012 ◽  
Vol 230 (3) ◽  
pp. 1060-1077 ◽  
Author(s):  
C.T. Chong ◽  
Theodore A. Slaman ◽  
Yue Yang
Keyword(s):  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Combinatorial Principles

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.

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.

NONSTANDARD MODELS IN RECURSION THEORY AND REVERSE MATHEMATICS

2014 ◽  
Vol 20 (2) ◽  
pp. 170-200 ◽  
Author(s):  
C. T. CHONG ◽  
WEI LI ◽  
YUE YANG
Keyword(s):  
Recursion Theory ◽  
Reverse Mathematics ◽  
Second Order ◽  
Ramsey's Theorem ◽  
Second Order Arithmetic ◽  
Ramsey’S Theorem ◽  
Nonstandard Models ◽  
Combinatorial Principles ◽  
Order Arithmetic ◽  
Effective Computability

AbstractWe give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

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.

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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