Amenable colorings
  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. 
Shimon Garti
math.LO