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.