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|ΰ kΎi}gεwjuModel theory and finite combinatoricsv

Model theory is a study of mathematical structures(algebraic structures and relational structures) from the viewpoint of logical formulas, roughly speaking, equations with boolean combinations and quantifiers. The classification of structures by model theoretic complexity, called stability, was started by S. Shelah's big project "Classification theory" in 1960's, and it has been studied with many generalizations and applications. Recently it was discovered that some of stability classes can be characterized with Ramsey classes, classes of finite structures with Ramsey property. In this talk, I'm going to introduce our recent work characterizing n-dependent theories, one of stability classes, by two way. One way uses a class of hyper graphs which is a Ramsey class, and the other is by proving a generalization of Sauer-Shelah's lemma which is a combinatorial lemma on VC-dimension of a family of sets. This is a joint work with A. Chernikov and D. Palacin.