### Abstract

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 TP_{1} or 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 SOPn (n-strong order property). Recently it has been proved that in any NSOP_{1} theory (i.e. a theory not having 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 SOP_{2} (which is equivalent to TP_{1}) and SOP_{1}. In addition, we study relationships between TP_{2} and Kim-forking, and show that a theory is supersimple iff there is no countably infinite Kim-forking chain.

Original language | English |
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Pages (from-to) | 287-301 |

Number of pages | 15 |

Journal | Fundamenta Mathematicae |

Volume | 249 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2020 Jan 1 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

*Fundamenta Mathematicae*,

*249*(3), 287-301. https://doi.org/10.4064/fm757-8-2019