Colouring of successor of regular again

TL;DR

This paper establishes a maximal version of a coloring property in set theory by proving a specific partition relation involving regular cardinals and their successors.

Contribution

It introduces a new maximal coloring property $Pr_1$ for successor cardinals under certain regularity conditions.

Findings

Proves $Pr_1(oldsymbol{ extcolor{red}{ ext{lambda}}},oldsymbol{ extcolor{red}{ ext{lambda}}},oldsymbol{ extcolor{red}{ ext{lambda}}},oldsymbol{ extcolor{red}{ heta}})$ for regular cardinals $ heta< ext{lambda}= ext{partial}^+$.

Establishes maximal coloring properties for successors of regular cardinals.

Contributes to the understanding of partition relations in set theory.

Abstract

We get a quite maximal version of the colouring property $Pr_1$ by proving $Pr_1(\lambda,\lambda,\lambda,\theta)$ when $\lambda = \partial^+, \partial > \theta$ are regular cardinals.