Are generically stable types stationary?

Question

Is every type that is generically stable over a model $M$, stationary over $M$? (Without any assumption on $T$.)

Some definitions:

A global type $p(x)$ is generically stable over $M$ if 1 and 2 below hold

1. $p(x)$ is definable over $M$, that is, for every formula $\varphi(x,y)\in L$ there is a formula $\vartheta(y)\in L(M)$ such that $\varphi(x,a)\in p$ iff $\vartheta(a)$.

2. $p(x)$ is finitely satisfiable in $M$, that is, for every formula $\varphi(x)\in p$ there is an $a\in M$ such that $\varphi(a)$.

I say that $p(x)$ is stationary over $M$ (I don't know if this is standard terminology) if $p(x)$ is the unique global type invariant over $M$ that extends $p_{\restriction M}(x)$.

Answer 1

In Proposition 1 (iv) on p. 4 of this paper, Pillay and Tanovic prove that in the context of an arbitrary theory $T$, if $p$ is a generically stable type over $A$, then $p$ is the unique global non-forking extension of $p|_A$. It follows that $p$ is the unique global $A$-invariant extension of $p|_A$, since $A$-invariant types do not fork over $A$.

The only problem is that the definition of "generically stable type" in that paper differs from yours. Pillay and Tanovic say that a global type $p$ is generically stable over $A$ if $p$ is $A$-invariant and for any Morley sequence $I$ in $p$ over $A$ and any formula $\phi(x)$ with parameters from the monster model, $\phi(I)$ is a finite or cofinite subset of $I$.

The Pillay-Tanovic definition implies yours (this is Proposition 1 (ii)) and is equivalent to yours if $T$ is NIP (see Theorem 2.29 in Simon's A Guide to NIP Theories), and the proof of this implication uses NIP in a serious way (but I don't have a counterexample in a theory with IP in mind, and I would like to see one!). So it seems like the Pillay-Tanovic definition is probably the right one to use outside the NIP setting.

Answer 2

This is too long for a comment, so I'm adding a followup answer, which is also motivated by the question raised at the end of Alex's answer. In particular, here is an example of a complete global type in an IP theory, which is definable and finitely satisfiable in any small model, but is not stationary over any small set. It's from the paper that I wrote with Kyle Gannon (Notre Dame) that Alex mentioned.

Let $T$ be the theory of the Fraisse limit of finite triangle-free graphs, in the language with a binary relation $E$. Let $\mathcal{U}$ be a monster model. There is a unique type $p\in S_1(\mathcal{U})$ that contains $\neg E(x,b)$ for all $b\in\mathcal{U}$. In particular, we necessarily have $x\neq b\in p$ for all $b\in\mathcal{U}$. By quantifier elimination, $p$ is definable over $\emptyset$.

Claim 1. If $M\prec \mathcal{U}$, then $p$ is finitely satisfiable in $M$.

Proof. We need to fix finite sets $A\subset\mathcal{U}$ and $B\subset\mathcal{U}$, and find some $c\in M$ such that $c\neq a$ for all $a\in A$, and $\neg E(c,b)$ for all $b\in B$. Suppose $|B|=n$. Let $\Gamma$ be a finite triangle-free graph whose chromatic number is greater than $n$. (It is a classical result that such graphs exist, e.g., via the Mycielskian operation.) We can assume that $\Gamma$ is an induced subgraph of $M$, and $\Gamma\cap A=\emptyset$. For a contradiction, suppose there is no $c\in \Gamma$ such that $\neg E(c,b)$ holds for all $b\in B$. Since $|B|=n$, we can partition $\Gamma$ into subsets $C_1,\ldots,C_m$ for some $m\leq n$, such that for all $i\leq m$ there is $b\in B$ satisfying $E(c,b)$ for all $c\in C_i$. Since $\mathcal{U}$ is triangle-free, each $C_i$ is an independent set (in the graph-theoretic sense). But this contradicts that $\Gamma$ has chromatic number greater than $n$.

Remark. In the paper, we actually prove that $p$ has the stronger property of being finitely approximated in any small model. For this proof, the fact that there are finite triangle-free graphs of arbitrarily large chromatic number is replaced by a stronger result of Erdos. In particular, for any $\epsilon>0$ and sufficiently large $n\geq 1$, there is a finite triangle-free graph of size $n$ such that any independent set of vertices has size at most $\epsilon n$.

Claim 2. $p$ is not stationary over any $B\subset\mathcal{U}$.

Proof. Fix $B\subset\mathcal{U}$. Let $c_*\in\mathcal{U}\backslash B$ be arbitrary such that $E(c_*,b_*)$ holds for some (fixed) $b_*\in B$. Let $C\subset\mathcal{U}$ be the set of realizations of tp$(c_*/B)$ in $\mathcal{U}$. Note that $C\cap B=\emptyset$. Also, $E(c,b_*)$ holds for all $c\in C$, which implies that $C$ is an independent set. So we have a consistent global type $q\in S_1(\mathcal{U})$ containing $x\neq b$ for all $b\in\mathcal{U}$, $\neg E(x,b)$ for all $b\in\mathcal{U}\backslash C$, and $E(x,c)$ for all $c\in C$. Then $q|_B=p|_B$, $q\neq p$, and $q$ is $B$-invariant by construction.

Remark. Via the work of Pillay and Tanovic mentioned in Alex's answer, it follows that $p$ is not generically stable over any small model. But this is also easy to see directly. In particular, if $M\prec\mathcal{U}$ and $(a_i)_{i<\omega}$ is a Morley sequence in $p|_M$, then $\{a_i:i<\omega\}$ is an independent set. So we can find some $b\in\mathcal{U}$ such that $\{i<\omega:E(a_i,b)\}$ is infinite and co-infinite.

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Since it's relevant, I thought I would add some remarks about Pillay and Tanovic's definition of generic stability. These observations help clarify the definition, and are useful for convincing one's self that this is the "right" definition. I'm going to state everything over small models.

Let $T$ be any complete theory and $\mathcal{U}$ a monster model.

Definition (Pillay & Tanovic). A type $p\in S_x(\mathcal{U})$ is generically stable over $M\prec\mathcal{U}$ if it is $M$-invariant and, for any ordinal $\alpha$, any Morley sequence $(a_i)_{i<\alpha}$ in $p|_M$, and any formula $\phi(x)$ over $\mathcal{U}$, the set $\{i<\alpha:\phi(a_i)\}$ is either finite or co-finite in $\alpha$.

So this is the same as what Alex wrote, except I am emphasizing that the Morley sequence is indexed by an arbitrary (infinite) ordinal. This is important because, for example, if $T$ is NIP then any $M$-invariant type satisfies the above definition for $\alpha=\omega$ (but it is not necessarily the case that every $M$-invariant type is generically stable).

In my opinion, this definition is very nicely clarified by the following proposition.

Proposition. Fix $M\prec\mathcal{U}$ and suppose $p\in S_x(\mathcal{U})$ is $M$-invariant. The following are equivalent.

1. $p$ is generically stable over $M$.
2. If $(a_i)_{i<\omega}$ is a Morley sequence in $p|_M$, then $$p=\big\{\phi(x)\in\mathcal{L}(\mathcal{U}):\{i<\omega:\phi(a_i)\}\text{ is cofinite in $\omega$}\big\}.$$
3. There does not exist an $\mathcal{L}$-formula $\phi(x;y)$, a Morley sequence $(a_i)_{i<\omega}$ in $p|_M$, and a sequence $(b_i)_{i<\omega}$ in $\mathcal{U}^y$ such that $\phi(a_i,b_j)$ holds if and only if $i\leq j$.

By definition, an $M$-invariant type $p$ is generically stable over $M$ if and only if the average type of any Morley sequence (of arbitrary infinite ordinal length) is complete. Condition 2 of the proposition is saying that it is enough to consider Morley sequences of length $\omega$, as long as one also has that the average type of the sequence coincides with $p$. In condition 3, if one replaces "Morley sequence" with "$M$-indiscernible sequence", then the result is the standard definition of a stable type. Overall I think this proposition is mostly folklore. These notes by Casanovas have some details on stable types and other relevant exercises. Section 2 of the paper with Gannon includes some details, as well as the fact that a global type is generically stable (over some small model) if and only if it is a frequency interpretation measure (see Chapter 7 of A Guide to NIP Theories).