More on tree properties
Volume 249 / 2020
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 ${\rm TP}_1$ 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.
