Indestructibility for Ramsey cardinals

This is a talk at the Minisymposium on ‘Large cardinals’, Joint Annual Conference of DMV and the ÖMG, University of Passau, September 28, 2021 (virtual).

Although Ramsey cardinals $\kappa$ were originally defined, as their name suggests, via a Ramsey-type coloring property, they can also be characterized by the existence of elementary embeddings on weak $\kappa$-models (transitive models $M\models{\rm ZFC}^-$ of size $\kappa$ with $\kappa\in M$). There is a standard toolbox of techniques for showing that a large cardinal characterized by the existence of some form of elementary embeddings is indestructible by a given forcing notion. The indestructibility arguments proceed by lifting an elementary embedding $j:M\to N$ with $\text{crit}(j)=\kappa$ of the type characterizing the large cardinal to an elementary embedding $j:M[G]\to N[H]$ in a forcing extension $V[G]$ by the desired forcing, and it is then verified that the lifted embedding is also of the type characterizing the large cardinal. The standard toolbox cannot be applied directly to elementary embeddings characterizing Ramsey cardinals (and other large cardinals in the vicinity of Ramsey cardinals) because the elementary embeddings $j:M\to N$ are on models that are not closed under $\lt\kappa$-sequences and standard arguments rely on the closure of the target model to build the $N$-generic filter $H$. Moreover, Ramsey-type embedding have special properties, which are not obviously preserved by the lifts. In this talk, I will provide techniques for lifting Ramsey-type elementary embeddings to a forcing extension. Using these techniques we can show that Ramsey cardinals are indestructible by the canonical forcing of the ${\rm GCH}$, that they can be made indestructible by the forcing ${\rm Add}(\kappa,\theta)$ (which implies that it is consistent for ${\rm GCH}$ to fail at a Ramsey cardinal), that they are indestructible by fast function forcing, and that they can be made indestructible by the club-shooting forcing, among other examples. This is joint work with Thomas Johnstone.