Complicated colorings, revisited

Abstract

In a paper from 1997, Shelah asked whether $P{r}_1({\lambda }^{+},{\lambda }^{+},{\lambda }^{+},\lambda )$ holds for every inaccessible cardinal $\lambda$. Here, we prove that an affirmative answer follows from $\square ({\lambda }^{+})$. Furthermore, we establish that for every pair $\chi <\kappa$ of regular uncountable cardinals, $\square (\kappa )$ implies $P{r}_1(\kappa ,\kappa ,\kappa ,\chi )$.