Combinatorial set theory

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The main focus of the proposal is the investigation of several types of infinite structure, using a range of mathematical tools. A secondary focus is the investigation of problems about very large finite structures, using certain infinite "limit" structures. Some of the problems to be studied are "test questions" intended to stimulate the development of new idea and methods.The topics of the proposal are mostly concerned with large cardinals, ZFC combinatorics (particularly of singular cardinals) and forcing. They include forcing axioms for successors of singulars, various forms of the tree property, connections between compactness properties, PCF theory and universality numbers. The last section of the proposal is concerned with problems arising in Razborov's theory of flag algebras.