On omega-categorical structures with few finite substructures

Pierre Simon

On omega-categorical structures with few finite substructures

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arxiv 1810.06531 v1 pith:LMOTMUR2 submitted 2018-10-15 math.LO math.CO

Pierre Simon This is my paper

classification math.LOmath.CO

keywords growthboundcaseexponentialfinitemerolaoptimalsequence

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We establish new results on the possible growth rates for the sequence (f_n) counting the number of orbits of a given oligomorphic group on unordered sets of size n. Macpherson showed that for primitive actions, the growth is at least exponential (if the sequence is not constant equal to 1). The best lower bound previously known for the base of the exponential was obtained by Merola. We establishing the optimal value of 2 in the case where the structure is unstable. This allows us to improve on Merola's bound and also obtain the optimal value for structures homogeneous in a finite relational language. Finally, we show that the study of sequences (f_n) of sub-exponential growth reduces to the omega-stable case.

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Structures with not too fast unlabelled growth
math.LO 2025-07 conditional novelty 7.0 of 10

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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