Continuous Ramsey theory in higher dimensions

• Applicant: Privatdozent Dr. Stefan Geschke
• Subject Area: Mathematics
• Term: from 2009 to 2013
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 156625774

The infinite Ramsey Theorem states that given a coloring of all n-element subsets of an infinite set with finitely many colors, there is an infinite set, all whose n-element subsets get the same color. A set with all its n-element subsets of the same color is homogeneous. This theorem fails when infinite is replaced by uncountable . If we restrict our attention to continuous colorings of two-element subsets of Polish spaces, an uncountable version of Ramsey s theorem can be proved, namely Galvin s theorem. For continuous colorings of n-element subsets of the Cantor space 2\omega a similar theorem is true, but one has to relax homogeneity to weak homogeneity. The following meta-mathematical statements is a strengthening of Galvin s theorem that can be formalized in first order logic: The set-theoretic universe has a generic extension in which for every Polish space X and every continous coloring of the two-element subsets of X, X is the union of fewer than 2x0 homogeneous sets. A similar theorem holds for continous colorings of the three-element subsets of the Cantor space, but again homogeneity has to be relaxed to weak homogeneity. One of the main goals of the proposed research is to generalize this result to continuous colorings of n-element subsets of the Cantor Space and to get a better understanding of the structure of continuous colorings of n-element subsets of Polish spaces in general. Such colorings play a role in convex geometry, the theory of Lipschitz functions and the combinatorics of infinite trees. Many results in continuous Ramsey theory are obtained through an interplay between finite and infinite combinatorics. We plan to publish the results of this research project together with previous results on continuous Ramsey theory in a monograph written jointly with Menachem Kojman, Ben Gurion University of the Negev.

• DFG Programme: Research Grants