Large cardinal ideals

Building on work of Holy, Lücke 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 $κ$ is weakly compact, many large cardinal ideals on $κ$ are nowhere $κ$-saturated. Lastly, we survey some recent consistency results concerning the weakly compact ideal as well as some recent results on the subtle, ineffable and $Π^1_1$-indescribable ideals on $P_κλ$, and we close with a list of open questions.

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