More on tree properties [article]

Enrique Casanovas, Byunghan Kim
2019 arXiv   pre-print
Tree properties are 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, transitivity, extension, local character, and type-amalgamation. Shelah also introduced SOP_n (n-strong order property). Recently it is proved that in any NSOP_1 theory (i.e. a theory not having SOP_1) holding nonforking existence,
more » ... orking also satisfies all the mentioned independence properties except base monotonicity (one direction of 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 obtain that a theory is supersimple iff there is no countably infinite Kim-forking chain.
arXiv:1902.08911v3 fatcat:hhu44653rzf4ras6ewldrcnwke