en>David Eppstein: source and expand

2014-01-18T23:49:40Z

source and expand

← Older revision		Revision as of 01:49, 19 January 2014
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			== Definition ==
			The diamond principle ◊ says that there exists a '''◊-sequence''', in other words sets ''A''<sub>α</sub>⊆α for α<ω<sub>1</sub> such that for any subset ''A'' of ω<sub>1</sub> the set of α with ''A''∩α = ''A''<sub>α</sub> is stationary in ω<sub>1</sub>.

			More generally, for a given [[cardinal number]] <math>\kappa</math> and a [[stationary set]] <math> S\subseteq\kappa </math>, the statement ◊<sub>''S''</sub> (sometimes written ◊(''S'') or ◊<sub>κ</sub>(''S'')) is the statement that there is a [[sequence]] <math>\langle A_\alpha: \alpha \in S \rangle </math> such that

			* each <math> A_\alpha \subseteq \alpha </math>
			* for every <math> A \subseteq \kappa, \{\alpha \in S: A \cap \alpha = A_\alpha\} </math> is stationary in <math>\kappa</math>

			The principle ◊<sub>ω<sub>1</sub></sub> is the same as ◊.

			== Properties and use ==

			{{harvtxt\|Jensen\|1972}} showed that the diamond principle ◊ implies the existence of [[Suslin tree]]s.
			He also showed that ◊ implies the [[continuum hypothesis\|CH]]. Also [[clubsuit\|♣]] + CH implies ◊, but [[saharon Shelah\|Shelah]] gave models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).

			{{harvtxt\|Akemann\|Weaver\|2004}} used ◊ to construct a [[C* algebra\|''C''<sup>*</sup>-algebra]] serving as a [[counterexample]] to [[Naimark's problem]].

			For all cardinals κ and stationary subsets ''S''&sube;κ<sup>+</sup>, ◊<sub>''S''</sub> holds in the [[constructible universe]]. Recently [[Saharon Shelah\|Shelah]] proved that for κ>ℵ<sub>0</sub>, ◊<sub>κ<sup>+</sup></sub> follows from <math>2^\kappa=\kappa^+</math>.

			==See also==
			*[[Statements true in L]]
			*[[Axiom of constructibility]]

			== References ==
			*{{Citation \| last1=Akemann \| first1=Charles \| last2=Weaver \| first2=Nik \| title=Consistency of a counterexample to Naimark's problem \| doi=10.1073/pnas.0401489101 \| mr=2057719 \| year=2004 \| journal=[[Proceedings of the National Academy of Sciences\|Proceedings of the National Academy of Sciences of the United States of America]] \| volume=101 \| issue=20 \| pages=7522–7525\|arxiv=math.OA/0312135}}
			*{{Citation \| last1=Jensen \| first1=R. Björn \| title=The fine structure of the constructible hierarchy \| doi=10.1016/0003-4843(72)90001-0 \| mr=0309729 \| year=1972 \| journal=Annals of Mathematical lLogic \| volume=4 \| pages=229–308}}
			* Assaf Rinot, ''Jensen's diamond principle and its relatives'', [http://papers.assafrinot.com/?num=s01 online]
			* S. Shelah: Whitehead groups may not be free, even assuming CH, II, ''Israel J. Math.'', '''35'''(1980), 257–285.

			[[Category:Set theory]]

en>David Eppstein: I tried searching for references but most "exact coloring" hits were for exact algorithms for the standard coloring problem. Tag as unreferenced.

2012-08-24T06:32:43Z

I tried searching for references but most "exact coloring" hits were for exact algorithms for the standard coloring problem. Tag as unreferenced.

New page

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Exact coloring - Revision history

en>David Eppstein: source and expand

en>David Eppstein: I tried searching for references but most "exact coloring" hits were for exact algorithms for the standard coloring problem. Tag as unreferenced.