PlanetMath: partitions less than cofinality

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partitions less than cofinality

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If $\lambda<\operatorname{cf}(\kappa)$ then $\kappa\rightarrow(\kappa)^1_\lambda$ .

This follows easily from the definition of cofinality. For any coloring $f:\kappa\rightarrow\lambda$ then define $g:\lambda\rightarrow\kappa+1$ by $g(\alpha)=\vert f^{-1}(\alpha)\vert$ . Then $\kappa=\sum_{\alpha<\lambda} g(\alpha)$ , and by the normal rules of cardinal arithmetic $\operatorname{sup}_{\alpha<\lambda} g(\alpha)=\kappa$ . Since $\lambda<\operatorname{cf}(\kappa)$ , there must be some $\alpha<\lambda$ such that $g(\alpha)=\kappa$ .

"partitions less than cofinality" is owned by Henry.

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See Also: arrows relation, arrows relation

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Cross-references: cardinal arithmetic, normal, coloring, cofinality
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This is version 3 of partitions less than cofinality, born on 2002-08-10, modified 2008-02-15.
Object id is 3287, canonical name is PartitionsLessThanCofinality.
Accessed 1480 times total.

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AMS MSC:

03E04 (Mathematical logic and foundations :: Set theory :: Ordered sets and their cofinalities; pcf theory)

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