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## Fundamenta Mathematicae

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## More on tree properties

### Tom 249 / 2020

Fundamenta Mathematicae 249 (2020), 287-301 MSC: Primary 03C45; Secondary 05C05. DOI: 10.4064/fm757-8-2019 Opublikowany online: 19 December 2019

#### Streszczenie

Tree properties were introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has ${\rm TP}_1$ or ${\rm TP}_2$. In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion as it satisfies symmetry, full transitivity, extension, local character, and type-amalgamation, over sets. Shelah also introduced ${\rm SOP}_n$ ($n$-strong order property). Recently it has been proved that in any ${\rm NSOP}_1$ theory (i.e. a theory not having ${\rm SOP}_1$) having nonforking existence, Kim-forking also satisfies all the above mentioned independence properties except base monotonicity (one direction of full transitivity). These results are the sources of motivation for this paper.

Mainly, we produce type-counting criteria for ${\rm SOP}_2$ (which is equivalent to ${\rm TP}_1$) and ${\rm SOP}_1$. In addition, we study relationships between ${\rm TP}_2$ and Kim-forking, and show that a theory is supersimple iff there is no countably infinite Kim-forking chain.

#### Autorzy

• Enrique CasanovasDepartament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via 585
08007 Barcelona, Spain
e-mail
• Byunghan KimDepartment of Mathematics
Yonsei University
50 Yonsei-ro Seodaemun-gu
Seoul 03722, Republic of Korea
e-mail

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