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Ramsey Theory

One indication of a bright future for reverse mathematics is that a number of outstanding recursion theorists are gradually getting drawn in. Of the researchers on Cholak's computability theory home page, several have published papers on reverse mathematics, including: Peter Cholak, Peter Clote, Rod Downey, Bill Gasarch, Jeff Hirst, Carl Jockusch, Steffen Lempp, Richard Shore, Ted Slaman. A recent major contribution is the Cholak/Jockusch/Slaman paper [7] on Ramsey's theorem for pairs.

Ramsey theory is a large subject with many interesting results in addition to the familiar Ramsey theorem. Moreover, there are important connections between Ramsey theory and dynamical systems theory, especially topological dynamics and ergodic theory. For a survey of the area, see the monograph of Graham/Rothschild/Spencer [12].

The purpose of this section is to mention some open problems regarding reverse mathematics and Ramsey theory. Such problems may be especially attractive, because Ramsey theory is a branch of mathematics where one could expect to find statements of high logical strength.


Stephen G Simpson