Large cardinal ideals

TL;DR

This paper characterizes large cardinal ideals, such as the Ramsey ideal, using generic elementary embeddings, compares different embedding approaches, and surveys recent consistency results and open questions in the field.

Contribution

It provides new characterizations of large cardinal ideals via generic embeddings and highlights differences between small and generic embedding approaches.

Findings

Characterizations of large cardinal ideals using generic embeddings.

Differences between small and generic embedding characterizations of subtle cardinals.

Many large cardinal ideals are nowhere -saturated when  is weakly compact.

Abstract

Building on work of Holy, L\"ucke and Njegomir \cite{MR3913154} on small embedding characterizations of large cardinals, we use some classical results of Baumgartner (see \cite{MR0384553} and \cite{MR0540770}), to give characterizations of several well-known large cardinal ideals, including the Ramsey ideal, in terms of generic elementary embeddings; we also point out some seemingly inherent differences between small embedding and generic embedding characterizations of subtle cardinals. Additionally, we present a simple and uniform proof which shows that, when $\kappa$ is weakly compact, many large cardinal ideals on $\kappa$ are nowhere $\kappa$-saturated. Lastly, we survey some recent consistency results concerning the weakly compact ideal as well as some recent results on the subtle, ineffable and $\Pi^1_1$-indescribable ideals on $P_\kappa\lambda$, and we close with a list of open questions.