Amenable colorings

TL;DR

This paper proves the consistency of certain advanced coloring relations in set theory assuming large cardinal axioms, extending known results to broader contexts.

Contribution

It establishes the consistency of complex coloring relations for regular cardinals under large cardinal assumptions, including a full amenable relation for specific cardinal pairs.

Findings

Proves the consistency of inom{\kappa^{++}}{\kappa^+} ightarrowinom{ au}{\kappa^+} for all au<\kappa^{++}

Establishes a full amenable relation for (\aleph_2,\aleph_1) with strongly closed collections

Assumes the existence of a huge cardinal above \kappa

Abstract

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.