Complicated colorings, revisited

Joint work with Jing Zhang.

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 $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}({\lambda }^{+})$.  Furthermore, we establish that for every pair $\chi <\kappa$ of regular uncountable cardinals, $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}(\kappa )$ implies $P{r}_1(\kappa ,\kappa ,\kappa ,\chi )$.

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