On $C^s_n(\kappa)$ and the Juhasz-Kunen question

TL;DR

This paper explores generalized combinatorial principles related to $C^s_n(\kappa)$ and addresses the Juhasz-Kunen question by establishing their independence and consistency results.

Contribution

It generalizes key combinatorial principles and proves that $C_n(\kappa)$ does not imply $C_{n+1}(\kappa)$, also showing the consistency of $C(\kappa)$ with the negation of $C^s(\kappa)$.

Findings

$C_n(\kappa)$ does not imply $C_{n+1}(\kappa)$ for $n>2$

Established the independence between $C^s(\kappa)$ and $C(\kappa)$

Provided consistency results for the principles

Abstract

We generalize the combinatorial principles $C_n(\kappa), C^s_n(\kappa)$ and $Princ(\kappa)$ introduced by various authors, and prove some of their properties and connections between them. We also answer a question asked by Juhasz-Kunen about the relation between these principles, by showing that $C_n(\kappa)$ does not imply $C_{n+1}(\kappa)$, for any $n>2$. We also show the consistency of $C(\kappa)+\neg C^s(\kappa).$