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Via two short proofs and three constructions, we show how to increase the model-theoretic precision of a widely used method for building ultrafilters. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a " bottleneck "… (More)

The first part of this paper is an expository overview of the au-thors' recent work on Keisler's order, a far-reaching program of understanding basic model-theoretic structure through the lens of regular ultrapowers. We motivate the problem and explain how this work connects model theory and set theory, leading to theorems on both sides. In the second part… (More)

We prove a regularity lemma with respect to arbitrary Keisler measures µ on V , ν on W where the bipartite graph (V, W, R) is definable in a saturated structure ¯ M and the formula R(x, y) is stable. The proof is rather quick, making use of local stability theory. The special case where (V, W, R) is pseudofinite, µ, ν are the counting measures, and ¯ M is… (More)

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