Complicated colorings, revisited

TL;DR

This paper explores the implications of certain combinatorial principles in set theory, proving that specific square principles imply particular coloring properties for uncountable cardinals, thus advancing understanding of their combinatorial structure.

Contribution

It demonstrates that the square principle $ox( heta)$ implies the coloring property $Pr_1( heta, heta, heta, u)$ for regular uncountable cardinals, extending prior results.

Findings

$ox( heta)$ implies $Pr_1( heta, heta, heta, u)$ for regular uncountable $ heta$

Affirmative answer to Shelah's question under $ox( heta)$

Connections established between square principles and coloring properties in set theory

Abstract

In a paper from 1997, Shelah asked whether $Pr_1(\lambda^+,\lambda^+,\lambda^+,\lambda)$ 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 $Pr_1(\kappa,\kappa,\kappa,\chi)$.