I'm wondering if there's any sort of Ramsey relation that allows for the tuples to be of arbitrary infinite size $\mu$? This $\mu$ is below some strongly compact cardinal, so I'm not worried about large cardinal hypotheses.



Is there Ramsey Theorem for infinitary tuples?
Infinite exponent partition relations are inconsistent with the axiom of choice, so in ZFC, this phenomenon does not exist, but nevertheless, in the context of $ZF+\neg AC$ there is a robust theory. See for example [Andres Caicedo's discussion](http://caicedoteaching.files.wordpress.com/2009/04/580-partition21.pdf), [this Kleinberg article](http://www.jstor.org/pss/2272066), and [the items in this Google search](http://www.google.com/search?q=infinite+exponent+partition+relations&rls=com.microsoft%3aen-us&ie=UTF-8&oe=UTF-8&startIndex=&startPage=1). 


