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 $g(\alpha)=\kappa$