On Cns(κ) and the Juhász–Kunen Question

Abstract

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