partitions less than cofinalityPartitionsLessThanCofinality2002-08-10 18:35:082008-02-15 20:39:13Resultcofinality% this is the default PlanetMath preamble. as your knowledge
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%\PMlinkescapeword{theory}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 $g(\alpha)=\kappa$.