Ramsey cardinal

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Ramsey cardinals were introduced by Erdős and Hajnal in [1]. A cardinal $\kappa$ is Ramsey if it has the partition property $\kappa\rightarrow (\kappa)^{\lt\omega}_2$ (see the weakly compact cardinals entry for notation meaning). They were named in honor of Frank Ramsey, whose Ramsey theorem for partition relations on $\omega$ motivated the generalizations to the larger infinite context. Ramsey cardinals is a robust large cardinal notion that is characterized model theoretically through the existence of $\kappa$-sized sets of indiscernibles for certain structures, as well as through the existence of well-founded ultrapowers for certain models of weak set theory.

Models with indiscernibles: Suppose $\mathcal A=(A,\ldots)$ is a model of a language $\mathcal L$ of size less than $\kappa$ whose universe $A$ contains $\kappa$ as a subset. Then $\kappa$ is Ramsey if and only if every such model $\mathcal A$ has a $\kappa$-sized set of indiscernibles $H\subseteq\kappa$, i.e. for every formula $\varphi(\overline x)$ of $\mathcal L$ and every pair of tuples $\overline \alpha$ and $\overline \beta$ in $H$, we have $\mathcal A\models\varphi (\overline \alpha)\leftrightarrow \varphi(\overline \beta)$.

Good sets of indiscernibles: Suppose $A\subseteq\kappa$, then $L_\kappa[A]$ denotes the $\kappa^{\text{th}}$-level of the universe constructible using a predicate for $A$. A set $I\subseteq\kappa$ is a good set of indiscernibles for the model $\langle L_\kappa[A],A\rangle$ if for all $\gamma\in I$,

  • $\langle L_\gamma[A\cap \gamma],A\cap \gamma\rangle\prec \langle L_\kappa[A], A\rangle$
  • $I\setminus\gamma$ is a set of indiscernibles for the model $\langle L_\kappa[A], A,\xi\rangle_{\xi\in\gamma}$.

A cardinal $\kappa$ is Ramsey if and only if for every $A\subseteq\kappa$, there is a good set of indiscernibles for the model $\langle L_\kappa[A], A\rangle$ of size $\kappa$.