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This week, Henry Towsner continued some model-theoretic preliminaries for the reading seminar of the Hrushovski paper, particularly regarding the behaviour of wide types, leading up to the main model-theoretic theorem (Theorem 3.4 of Hrushovski) which in turn implies the various combinatorial applications (such as Corollary 1.2 of Hrushovski). Henry’s notes can be found here.

A key theme here is the phenomenon that any pair of large sets contained inside a definable set of finite measure (such as ${X \cdot X^{-1}}$) must intersect if they are sufficiently “generic”; the notion of a wide type is designed, in part, to capture this notion of genericity.