Effectiveness for infinite variable words and the Dual Ramsey Theorem
نویسندگان
چکیده
منابع مشابه
Effectiveness for infinite variable words and the Dual Ramsey Theorem
We examine the Dual Ramsey Theorem and two related combinatorial principles VW(k, l) and OVW(k, l) from the perspectives of reverse mathematics and effective mathematics. We give a statement of the Dual Ramsey Theorem for open colorings in second order arithmetic and formalize work of Carlson and Simpson [1] to show that this statement implies ACA0 over RCA0. We show that neither VW(2, 2) nor O...
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The classical Ramsey theorem was generalized in two major ways: to the dual Ramsey theorem, by Graham and Rothschild, and to Ramsey theorems for trees, initially by Deuber and Leeb. Bringing these two lines of thought together, we prove the dual Ramsey theorem for trees. Galois connections between partial orders are used in formulating this theorem, while the abstract approach to Ramsey theory,...
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The infinite Ramsey theorem is known to be equivalent to the statement ‘for every set X and natural number n, the n-th Turing jump of X exists’, over RCA0 due to results of Jockusch [5]. By subjecting the theory RCA0 augmented by the latter statement to an ordinal analysis, we give a direct proof of the fact that the infinite Ramsey theorem has proof-theoretic strength εω. The upper bound is ob...
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In 2012 M. Sokić proved that the that the class of all finite permutations has the Ramsey property. Using different strategies the same result was then reproved in 2013 by J. Böttcher and J. Foniok, in 2014 by M. Bodirsky and in 2015 yet another proof was provided by M. Sokić. Using the categorical reinterpretation of the Ramsey property in this paper we prove that the class of all finite permu...
متن کاملNotes on the Dual Ramsey Theorem
The goal of these notes is to give a proof of the Dual Ramsey Theorem. This theorem was first proved in “A Dual Form of Ramsey’s Theorem” by Tim Carlson and Steve Simpson, Advances in Mathematics, 1984. The proof presented here is the same proof given in the paper and follows much of the same notation and terminology. The main change is that we will explicitly use the notion of “infinite variab...
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ژورنال
عنوان ژورنال: Archive for Mathematical Logic
سال: 2004
ISSN: 0933-5846,1432-0665
DOI: 10.1007/s00153-004-0216-4