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NIP omega-categorical structures: the rank 1 case
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We classify primitive, rank 1, omega-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely many such structures and that all are built out of finitely many linear orders interacting in a restricted number of ways. As an example of application, we deduce the classification of primitive structures homogeneous in a language consisting of n linear orders as well as all reducts of such structures.
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Structures with not too fast unlabelled growth
A complete classification of ω-categorical structures with unlabelled growth below 2^n/p(n), confirming Thomas' conjecture and giving optimal growth gaps for this class.
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