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.

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    $\begingroup$ There are some generalizations of the Galvin-Prikry Ramsey theorem to higher cardinals. The experts here can no doubt point to them. One source I found with a quick google search is R.J. Watro's paper, "On partitioning the infinite subsets of large cardinals", J. Symbolic Logic 49 no. 2, June, 1984. $\endgroup$ – Bill Johnson Jun 11 '11 at 4:57

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, this Kleinberg article, and the items in this Google search.


As emphasized in Joel Hamkins' answer, the generalization of Ramsey's theorem for infinite (unordered) tuples contradicts the axiom of choice [Erdős-Hajnal, 1966], and is a line of investigation that has close ties to large cardinals.

The classical Erdős-Hajnal proof uses the axiom of choice - in the guise of a well-ordering of the power set of $\Bbb {N}$ - to construct a "wild" coloring $C$ of infinite subsets $[\Bbb{N}]^\omega$ of $\Bbb{N}$ into two colors such that there is no infinite monochromatic set for $C$.

In contrast, Galvin and Prikry showed that for Borel colorings $C$ of $[\Bbb{N}]^\omega$, an infinite monochromatic subset for $C$ always exists. Silver then extended this result to analytic colorings. Note that $[\Bbb{N}]^\omega$ inherits a natural topology from $P(\Bbb{N})$, which is itself topologized via an identification with the product space $2^\Bbb{N}$.

The Galvin-Prikry paper appeared in 1973, but that of Silver appeared in 1970 (this is not a typo!). This work was simplified and extended by Ellentuck in 1974.

The metamathematics of Ramsey theory, including Galvin-Prikry type theorems, has been vigorously investigated in reverse mathematics.


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