A refinement of the Ramsey hierarchy via indescribability

Abstract

A subset $S$ of a cardinal $\kappa$ is Ramsey if for every function $f:[S]^{<\omega}\to \kappa$ with $f(a)<\min a$ for all $a\in[S]^{<\omega}$, there is a set $H\subseteq S$ of cardinality $\kappa$ which is \emph{homogeneous} for $f$, meaning that $f\upharpoonright[H]^n$ is constant for each $n<\omega$. Baumgartner proved \cite{MR0384553} that if $\kappa$ is a Ramsey cardinal, then the collection of non-Ramsey subsets of $\kappa$ is a normal ideal on $\kappa$. Sharpe and Welch \cite{MR2817562}, and independently Bagaria \cite{MR3894041}, extended the notion of $\Pi^1_n$-indescribability where $n<\omega$ to that of $\Pi^1_\xi$-indescribability where $\xi\geq\omega$. We study large cardinal properties and ideals which result from Ramseyness properties in which homogeneous sets are demanded to be $\Pi^1_\xi$-indescribable. By iterating Feng's Ramsey operator \cite{MR1077260} on the various $\Pi^1_\xi$-indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide a complete account of the containment relationships between the resulting ideals and show that the corresponding large cardinal properties yield a strict linear refinement of Feng's original Ramsey hierarchy. We also show that, given any ordinals $\beta_0,\beta_1<\kappa$ the increasing chains of ideals obtained by iterating the Ramsey operator on the $\Pi^1_{\beta_0}$-indescribability ideal and the $\Pi^1_{\beta_1}$-indescribability ideal respectively, are eventually equal; moreover, we identify the least degree of Ramseyness at which this equality occurs. As an application of our results we show that one can characterize our new large cardinal notions and the corresponding ideals in terms of generic elementary embeddings; as a special case this yields generic embedding characterizations of $\Pi^1_\xi$-indescribability and Ramseyness.