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Suppose $M$ is a first-order structure which is unstable. If necessary, assume it is $\aleph_0$-saturated (or more, but I don't think it matters beyond that).

Are there any interesting criteria for determining whether $M$ has an unstable $\aleph_0$-categorical reduct? Was this problem considered? (Or any related problem, with "unstable" replaced by some other relevant adjective.)

(By reduct I mean a structure obtained from $M$ by expanding it by definitions, possibly allowing parameters, and then forgetting some language; for this problem, I think we can equivalently replace "reduct" by "a structure interpreted in $M$".)

Note that the question trivialises when we drop "unstable", since the trivial reduct is $\aleph_0$-categorical.

Note also that it is enough to consider reducts with only one relational symbol, since that's all we need to witness unstability. Thus, the question would be resolved if we knew how to determine whether, given a formula $\varphi(x,y)$, the reduct to $\varphi$ is $\aleph_0$-categorical.

I guess this question does not have a trivial answer for arbitrary $M$, since if this was true (that every unstable structure admits an unstable $\aleph_0$-categorical reduct), it would follow (by the work of Pierre Simon) that the conjecture (of Shelah IIRC) that every unstable NIP theory defines an infinite linear order is true, which to my knowledge remains open (and I think I remember Pierre saying that he thinks it's probably not true).

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    $\begingroup$ I doubt that any infinite discrete linear order has an unstable $\aleph_0$-categorical reduct. $\endgroup$
    – Alex Kruckman
    Commented Sep 23 at 23:43
  • $\begingroup$ @AlexKruckman: You're probably right. I guess this should not be too hard to show via o-minimality. $\endgroup$
    – tomasz
    Commented Sep 24 at 12:34
  • $\begingroup$ Do we have any idea whether it's necessary to consider reducts with parameters? Both stability and $\aleph_0$-categoricity are invariant under adding (finitely many) parameters (i.e., $T$ satisfies one of these conditions iff $T_{\bar{a}}$ satisfies it as well). $\endgroup$
    – James E Hanson
    Commented Sep 24 at 16:29
  • $\begingroup$ @JamesEHanson: It's not clear to me either way. For a given formula $\varphi(x,y,z)$, I can certainly think of a scenario where $\varphi(x,y,z)$ allows us to have infinitely many types in the $z$ sort, but $\varphi(x,y,a)$ is $\aleph_0$-categorical for a suitably chosen $a$. But perhaps an altogether different formula without parameters would work... $\endgroup$
    – tomasz
    Commented Sep 24 at 18:52

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