A dual form of Ramsey's Theorem

Carlson, Timothy J.; Simpson, Stephen G.

doi:10.1016/0001-8708(84)90026-4

Cited by 120 publications

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“…We let ODRT(k, l) stand for this theorem with fixed values of k and l. Formalizing a proof given by Carlson and Simpson [1], we show that ODRT(k + 1, l) implies RT(k, l) over RCA 0 . From the discussion above, this immediately shows that for k ≥ 3 and l ≥ 2, ODRT(k, l) is not provable in WKL 0 .…”

Section: Introductionmentioning

confidence: 61%

“…In this article, we perform a similar (though less comprehensive) analysis for several variations of Ramsey's Theorem which were introduced by Carlson and Simpson [1] and later used by Simpson [6] to study initial segment constructions in degree theory. We begin with a classical statement of the Dual Ramsey Theorem as proved in [1].…”

Section: Introductionmentioning

confidence: 99%

“…We begin with a classical statement of the Dual Ramsey Theorem as proved in [1]. Let (N) N be the set of all partitions of N into infinitely many pieces and let (N) k be the set of all partitions of N into k pieces.…”

Section: Introductionmentioning

confidence: 99%

“…Friedman and Simpson [3] also asked for an analysis of Thereom 6.3 from Carlson and Simpson [1], a strengthening of one of the crucial lemmas in the proof of the Dual Ramsey Theorem. We state this principal in two different forms and with slightly different terminology than [3].…”

Section: Introductionmentioning

confidence: 99%

“…(In this situation, we say that W is a homogeneous word for c.) We let OVW(k, l) be the same statement except that we require W to be an infinite ordered variable word. There is a proof of OVW(k, l) for all k and l in Carlson and Simpson [1,Theorem 6.3]. Friedman and Simpson [3] asked for an analysis of OVW(k, l) and in particular, they asked if this statement is effectively true.…”

Section: Introductionmentioning

confidence: 99%

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Effectiveness for infinite variable words and the Dual Ramsey Theorem

Miller

2004

Archive for Mathematical Logic

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Abstract. 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 ACA 0 over RCA 0 . We show that neither VW(2, 2) nor OVW(2, 2) is provable in WKL 0 . These results give partial answers to questions posed by Friedman and Simpson [3].

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Section: Introductionmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Effectiveness for infinite variable words and the Dual Ramsey Theorem

Miller

2004

Archive for Mathematical Logic

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Partitions with no Coarsenings of Higher Degree

Brackin¹

1989

Mathematical Logic Qtrly

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There exist infinite subsets of the natural numbers with no subsets of higher Turing degree. This was established by R. I . SOARE [3] in a construction using the GalvinPrikry theorem. S. G. SIMPSON and T. J. CARLSON [2] proved a "dual" form of the Galvin-Prikry theorem and its extension by E. ELLENTUCK that applies to partitions and their coarsenings rather than to sets and their subsets. SIMPSON conjectured that the "dual" form of SOARE'S result was also true; i.e. that there exist partitions of the natural numbers into infinite numbers of classes with no infinite coarsenings of higher degree. A proof of SIMPSON'S conjecture follows. It employs a construction that mirrors SOARE'S, but uses the dual Ellentuck theorem in place of the Galvin-Prikry theorem.The following notation and terminology are drawn, for the most part, from SIMPSON and CARLSON [a]. Let N denote the natural numbers, including zero, and identify each n E N with the set of smaller natural numbers, i.e. n = (0, . . ., n -l}. If S is either N or some n E N, let a partition of S be a collection of pairwise disjoint, nonempty subsets of S whose union is S. Call the elements of such a partition its blocks, and call the smallest element of each block a leader. If X and Y are partitions of S , say that Y is coarser than X , or that Y i s a coarsening of X , if every block in X is a subset of some block in Y . Call elements of the same block equivalent. Call a partition of N infinite if it contains infinitely many blocks.If X is a partition of N, n E N, and s is a partition of n, say that s i s a segment of X , written s X , if s is the partition of n induced by X , i.e. if s = X [ n ] = { X A n : X E X } -{ O } .If s is a partition of n, call n the length of s, and denote it by lh(s); note that all coarsenings of s have the same length.Let (s, X ) + denote the set of all coarsenings of X , finite or infinite, for which s is a segment.The subsets of (N)" of the form (s, X ) are called dual Ellentuck neighborhoods, and the dual Ellentuck topology on (N)" is the topology whose basic open sets are the dual Ellentuck neighborhoods. The standard topology on (N)" is the topology it inherits l ) This material was part of the author's doctoral dissertation at the Pennsylvania State University [I]. He wishes to thank his dissertation advisor, STEPHEN G. SIMPSON.

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On Shattering, Splitting and Reaping Partitions

Halbeisen

1998

Mathematical Logic Qtrly

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In this article we investigate the dual-shattering cardinal 9, the dual-splitting cardinal 6 and the dual-reaping cardinal B, which are dualizations of the well-known cardinals fj (the shattering cardinal, also known as the distributivity number of P(w)/fin), 5 (the splitting number) and t (the reaping number). Using some properties of the ideal 3 of nowhere dual-Ramsey sets, which is an ideal over the set of partitions of w , we show that add(3) = cov(3) = 9. With this result we can show that 9 > w1 is consistent with ZFC and as a corollary we get the relative consistency of 9 > t, where t is the tower number. Concerning 6 we show that cov(M) 5 6 (where M is the ideal of the meager sets). For the dual-reaping cardinal B we get p 5 % 5 t (where p is the pseudo-intersection number) and for a modified dual-reaping number %' we get %' 5 3 (where 3 is the dominating number). As a consistency result we get % < cov(M).Mathematics Subject Classification: 03E05, 03E35, 03C25, 04A20, 05A18. The set of partitionsA partial partition X (of w ) is a family of pairwise disjoint, nonempty sets, such that dom(X) := u X w . The elements of a partial partition X are called the blocks of X and Min(X) denotes the set of the least elements of the blocks of X . If dom(X) = w , then X is called a partition. ( w } is the partition containing only one block. The set of all partitions containing infinitely (resp. finitely) many blocks is denoted by ( w )~ (resp. ( w ) < " ) . By (w)" we denote the set of all infinite partitions such that at least one block is infinite. The set of all partial partitions with dom(X) E w is denoted Let X I , X 2 be two partial partitions. We say that X1 is coarser than X2, or that X2 is finer than X I , and write X1 C. X 2 , if for all blocks b E X1 the set bndom(X2) is the union of some sets bi ndom(X1), where each bi is a block of X 2 . (Note that if X1 is coarser than X 2 and dom(X1) dom(XZ), then X1 is in a natural way also contained in X z .) Let X1 n Xz denotes the finest partial partition which is coarser than X I and X 2 such that dom(X1 fl X z ) = dom(X1) U dom(X2). Similarly X1 U X 2 denotes the coarsest partial partition which is finer than XI and X z such that dom(X1 U X , ) = dom(X1) tl dom(X2). by (W.')The author wishes to thank the Swiss Nationalfonds for supporting him. 2)

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A dual form of Ramsey's Theorem

Cited by 120 publications

References 18 publications

Effectiveness for infinite variable words and the Dual Ramsey Theorem

Effectiveness for infinite variable words and the Dual Ramsey Theorem

Partitions with no Coarsenings of Higher Degree

On Shattering, Splitting and Reaping Partitions

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