Amenable colorings

Abstract

Let $\kappa$ be any regular cardinal. Assuming the existence of a huge cardinal above $\kappa$, we prove the consistency of $\left(\genfrac{}{}{0ex}{}{{\kappa }^{++}}{{\kappa }^{+}}\right)\to \left(\genfrac{}{}{0ex}{}{\tau }{{\kappa }^{+}}\right)$ 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.

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