# partitions less than cofinality

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 $g(\alpha)=\kappa$.

Title partitions less than cofinality PartitionsLessThanCofinality 2013-03-22 12:55:56 2013-03-22 12:55:56 Henry (455) Henry (455) 6 Henry (455) Result msc 03E04 Arrowsrelation ArrowsRelation