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Local character of Kim-independence


Authors: Itay Kaplan, Nicholas Ramsey and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 147 (2019), 1719-1732
MSC (2010): Primary 03C45, 03C55, 03C80
DOI: https://doi.org/10.1090/proc/14305
Published electronically: January 8, 2019
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Abstract: We show that $ \mathrm {NSOP}_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $ T$ is $ \mathrm {NSOP}_{1}$, $ M\models T$, and $ p$ is a complete type over $ M$, then the collection of elementary substructures of size $ \left \vert T\right \vert$ over which $ p$ does not Kim-fork is a club of $ \left [M\right ]^{\left \vert T\right \vert}$ and that this characterizes $ \mathrm {NSOP}_{1}$.

We also present a new phenomenon we call dual local-character for Kim-independence in $ \mathrm {NSOP}_{1}$ theories.


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Additional Information

Itay Kaplan
Affiliation: Einstein Institute of Mathematics, Hebrew University of Jerusalem, Edmond J. Safra Campus Givat Ram, 91904 Jerusalem, Israel

Nicholas Ramsey
Affiliation: Department of Mathematics, University of California, Berkeley, 970 Evans Hall 3840, Berkeley, California 94720
Address at time of publication: Department of Mathematics, University of California, Los Angeles, Math Sciences Building 6363, Los Angeles, California 90095

Saharon Shelah
Affiliation: Einstein Institute of Mathematics, Hebrew University of Jerusalem, Edmond J. Safra Campus Givat Ram, 91904 Jerusalem, Israel

DOI: https://doi.org/10.1090/proc/14305
Received by editor(s): July 14, 2017
Received by editor(s) in revised form: February 12, 2018, and June 19, 2018
Published electronically: January 8, 2019
Additional Notes: The first author would like to thank the Israel Science Foundation for partial support of this research (Grant no. 1533/14).
The third author was partially supported by European Research Council grant 338821, number 1118.
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2019 American Mathematical Society