Properties of independence in ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_3$ theories

We prove some results about the theory of independence in ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_3$ theories that do not hold in ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_4$ theories. We generalize Chernikov’s work on simple and co-simple types in ${\mathrm{N}\mathrm{T}\mathrm{P}}_2$ theories to types with ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_1$ induced structure in $\mathrm{N}$-$\omega$-${\mathrm{D}\mathrm{C}\mathrm{T}\mathrm{P}}_2$ and ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_3$ 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 ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_3$ theories whose internal structure is ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_1$. Additionally, we show that in ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_3$ theories with symmetric Conant-independence, finitely satisfiable types satisfy an independence theorem similar to one conjectured by Simon for invariant types in ${\mathrm{N}\mathrm{T}\mathrm{P}}_2$ theories, and give generalizations of this result to invariant and Kim-nonforking types.