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Stable regularity for relational structures

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arxiv 1712.09305 v2 pith:DOLTQFBF submitted 2017-12-26 math.LO math.CO

Nathanael Ackerman , Cameron Freer , Rehana Patel This is my paper · ORCID
classification math.LOmath.CO
keywords finiteregularitystructurespartitionrelationalstabilitystableaccuracy
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abstract

We generalize the stable graph regularity lemma of Malliaris and Shelah to the case of finite structures in finite relational languages, e.g., finite hypergraphs. We show that under the model-theoretic assumption of stability, such a structure has an equitable regularity partition of size polynomial in the reciprocal of the desired accuracy, and such that for each $k$-ary relation and $k$-tuple of parts of the partition, the density is close to either 0 or 1. In addition, we provide regularity results for finite and Borel structures that satisfy a weaker notion that we call almost stability.

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  1. Quantitative analytic stable regularity

    math.LO 2026-07 conditional novelty 7.0 of 10

    Stable real-valued functions, defined by omitting ladders, admit definable partitions and equipartitions with polynomial-in-1/ε many parts such that every pair is almost constant.

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