Properties of independence in ${\mathrm{NSOP}}_3$ theories

Abstract

We prove some results about the theory of independence in ${\mathrm{NSOP}}_3$ theories that do not hold in ${\mathrm{NSOP}}_4$ theories. We generalize Chernikov's work on simple and co-simple types in ${\mathrm{NTP}}_2$ theories to types with ${\mathrm{NSOP}}_1$ induced structure in $\mathrm{N}$-$\omega$-${\mathrm{DCTP}}_2$ and ${\mathrm{NSOP}}_3$ theories, and give an interpretation of our arguments and those of Chernikov in terms of the characteristic sequences introduced by Malliaris. We then prove an extension of the independence theorem to types in ${\mathrm{NSOP}}_3$ theories whose internal structure is ${\mathrm{NSOP}}_1$. Additionally, we show that in ${\mathrm{NSOP}}_3$ theories with symmetric Conant-independence, finitely satisfiable types satisfy an independence theorem similar to one conjectured by Simon for invariant types in ${\mathrm{NTP}}_2$ theories, and give generalizations of this result to invariant and Kim-nonforking types.

arXiv comment29 pages

References in corpus (6)

• Generic expansion and Skolemization in NSOP${}_1$ theories
• Indiscernibles, EM-types, and Ramsey Classes of Trees
• Edge distribution and density in the characteristic sequence
• Dividing Lines between Positive Theories
• On amalgamation in NTP2 theories and generically simple generics
• Some Remarks on Kim-dividing in NATP Theories