Is there Ramsey Theorem for infinitary tuples? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:27:40Z http://mathoverflow.net/feeds/question/67483 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67483/is-there-ramsey-theorem-for-infinitary-tuples Is there Ramsey Theorem for infinitary tuples? Will 2011-06-11T02:38:13Z 2011-06-11T18:59:16Z <p>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.</p> http://mathoverflow.net/questions/67483/is-there-ramsey-theorem-for-infinitary-tuples/67484#67484 Answer by Joel David Hamkins for Is there Ramsey Theorem for infinitary tuples? Joel David Hamkins 2011-06-11T02:42:07Z 2011-06-11T02:49:39Z <p>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 <a href="http://caicedoteaching.files.wordpress.com/2009/04/580-partition21.pdf" rel="nofollow">Andres Caicedo's discussion</a>, <a href="http://www.jstor.org/pss/2272066" rel="nofollow">this Kleinberg article</a>, and <a href="http://www.google.com/search?q=infinite+exponent+partition+relations&amp;rls=com.microsoft%3aen-us&amp;ie=UTF-8&amp;oe=UTF-8&amp;startIndex=&amp;startPage=1" rel="nofollow">the items in this Google search</a>. </p> http://mathoverflow.net/questions/67483/is-there-ramsey-theorem-for-infinitary-tuples/67490#67490 Answer by Ali Enayat for Is there Ramsey Theorem for infinitary tuples? Ali Enayat 2011-06-11T06:19:35Z 2011-06-11T06:19:35Z <p>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.</p> <p>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 $C$.</p> <p><strong>In contrast, Galvin and Prikry showed that for <em>Borel</em> colorings $C$ of $[\Bbb{N}]^\omega$, an infinite monochromatic subset for $C$ always exists. Silver then extended this result to <em>analytic</em> 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 $2^\Bbb{N}$.</strong></p> <p>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. </p> <p>The metamathematics of Ramsey theory, including Galvin-Prikry type theorems, has been vigorously investigated in <a href="http://en.wikipedia.org/wiki/Reverse_mathematics" rel="nofollow">reverse mathematics</a>.</p>