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Several types of infinite cardinals are easily defined in terms of partition relations. For instance, if $\kappa > \omega$ then

  • $\kappa$ is weakly compact if $\kappa$ satisfies $\kappa\to(\kappa)^2_2$
  • $\kappa$ is Ramsey if $\kappa$ satisfies $\kappa\to(\kappa)^{<\omega}_2$

I have not seen definitions of cardinals larger than Ramsey cardinals in terms of similar partition relations. The definitions of measurable cardinals (and higher) typically make use of other notions such as critical points of elementary embeddings, ultrafilters with particular properties, or even more intricate ideas.

Can partitions relations of the form $\kappa\to(\lambda)^{\mu}_{\nu}$ be used to define larger cardinals than Ramsey cardinals?

If yes, is there an upper-limit? (where?) If no, does that mean that as soon as we start using elementary-embeddings, etc., we've gone beyond the scope of what partition relations can capture?

Thanks in advance for any answers, comments or references.

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up vote 1 down vote accepted

The reason you have no seen such characterization is that $\kappa\to(\kappa)^\omega_2$ is inconsistent with the axiom of choice.

One hint towards the proof would be to well-order $[\kappa]^\omega$ by some $\prec$ and consider the coloring $F(x)=1$ if and only if $\forall y\subsetneq x:y\nprec x$. Now prove that a homogeneous set of color $0$ would correspond to a decreasing sequence of ordinals, and it would be impossible to have a homogeneous set of color $1$ due to the definition of the coloring.

On the other hand, in some models without the axiom of choice $\omega_1\to(\omega_1)^{\omega_1}_2$, but in such models $\omega_1$ carries a $\sigma$-complete ultrafilter (i.e. it is a measurable cardinal).

There are "finer" notions of partition-defined cardinals, e.g. Erdős cardinals which are weaker than Ramsey cardinals in general. Both Kanamori The Higher Infinite as well Jech's Set Theory, $3$rd millennium edition have chapters dedicated to partition principles, including the aforementioned results and cardinals, and both are considered the standard books for these topics, I believe.

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