  Let $\kappa$ be any regular cardinal. Assuming the existence of a huge
cardinal above $\kappa$, we prove the consistency of
$\binom{\kappa^{++}}{\kappa^+}\rightarrow\binom{\tau}{\kappa^+}$ for every
ordinal $\tau<\kappa^{++}$. Likewise, we prove a full amenable relation for
$(\aleph_2,\aleph_1)$ with respect to collections which are strongly closed
under countable intersections.

Logic