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
Canonical name	PartitionsLessThanCofinality
Date of creation	2013-03-22 12:55:56
Last modified on	2013-03-22 12:55:56
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	6
Author	Henry (455)
Entry type	Result
Classification	msc 03E04
Related topic	Arrowsrelation
Related topic	ArrowsRelation