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'partitions less than cofinality'
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| Title of object: |
partitions less than cofinality |
| Canonical Name: |
PartitionsLessThanCofinality |
| Type: |
Result |
| Created on: |
2002-08-10 18:35:08 |
| Modified on: |
2007-06-17 14:43:55 |
| Classification: |
msc:03E04 |
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Content:
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 $g(\alpha)=|f^{-1}(\alpha)|$. 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 $g(\alpha)=\kappa$. |
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