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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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. |
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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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