Ramsey theory of homogeneous structures: current trends and open problems
Natasha Dobrinen
University of Denver, Department of Mathematics, 2390 S. York St., Denver, CO 80210, USA; current address: University of Notre Dame, Department of Mathematics, 255 Hurley Building, Notre Dame, IN 46556, USA
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Abstract
This article highlights historical achievements in the partition theory of countable homogeneous relational structures, and presents recent work, current trends, and open problems. Exciting recent developments include new methods involving logic, topological Ramsey spaces, and category theory. The paper concentrates on big Ramsey degrees, presenting their essential structure where known and outlining areas for further development. Cognate areas, including infinite dimensional Ramsey theory of homogeneous structures and partition theory of uncountable structures, are also discussed.