Preservation of additional tameness properties for separable pairs

Investigate whether the preservation theorem for stability and NIP proved for the theory of separable pairs of fields also holds for other tameness notions, in particular simplicity, NTP2, NIPn, and NSOPn.

Background

The paper proves that, for a complete theory T of fields of characteristic p, stability and NIP pass from T to the theory SCF_T of separable pairs whose large field is separably closed. The authors then discuss whether analogous preservation results hold for further model-theoretic tameness properties. They note that methods used elsewhere for algebraically closed pairs might extend to the separable setting, but they do not establish any such extensions.

The unresolved question explicitly concerns simplicity, NTP2, NIPn, and NSOPn, with n ranging over the relevant natural numbers. It is included because the paper leaves these preservation problems open rather than answering them.

References

We suspect that various other tameness notions are preserved as well. In particular, it seems plausible that the methods from employed to show preservation of simplicity and NSOP$_1$ for $ACF_T$ may be transferred to the case of $SCF_T$. We did not investigate this further and leave it as a question. Does the preservation result in Theorem~\ref{T:preserve-tame} hold for other tameness notions, in particular for simplicity, NTP$_2$, NIP$_n$ and/or NSOP$_n$ (where $n\in$)?

— On separable pairs of fields and beautiful pairs of separably closed valued fields  (2610.18753 - He et al., 13 Oct 2026) in Section 5, immediately following Theorem 5.12 (the theorem labeled T:preserve-tame)