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The stable regularity lemma revisited


Authors: Maryanthe Malliaris and Anand Pillay
Journal: Proc. Amer. Math. Soc. 144 (2016), 1761-1765
MSC (2010): Primary 03C45, 03C98, 05C75
DOI: https://doi.org/10.1090/proc/12870
Published electronically: September 9, 2015
MathSciNet review: 3451251
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Abstract: We prove a regularity lemma with respect to arbitrary Keisler measures $ \mu $ on $ V$, $ \nu $ on $ W$ where the bipartite graph $ (V,W,R)$ is definable in a saturated structure $ {\bar 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, $ \mu $, $ \nu $ are the counting measures, and $ {\bar M}$ is suitably chosen (for example a nonstandard model of set theory), yields the stable regularity theorem of a work by the first author and S. Shelah, though without explicit bounds or equitability.


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Additional Information

Maryanthe Malliaris
Affiliation: Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637
Email: mem@math.uchicago.edu

Anand Pillay
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: apilla@nd.edu

DOI: https://doi.org/10.1090/proc/12870
Received by editor(s): May 1, 2015
Published electronically: September 9, 2015
Additional Notes: The first author was partially supported by NSF grant DMS-1300634 and a Sloan fellowship
The second author was partially supported by NSF grant DMS-1360702
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2015 American Mathematical Society