Fact 1.2 [Reference SimonSim20, Lemma 2.2].

Let $\pi (x)$ be a partial A-invariant type. Then $\pi $ is ind-definable over A if and only if the set $X=\{(a,\overline {b}) : \overline {b}\in \mathbb {M}^{\omega }, a\vDash \pi |A\bar b\}$ is type-definable over A.

Fact 1.5 [Reference SimonSim20, Lemma 2.4].

Let $\pi $ be a partial type ind-definable over A. Let $a\vDash \pi |A$ and b such that $\mathrm {tp}(b/Aa)$ is finitely satisfiable in A. Then $a\vDash \pi |Ab$ .

Fact 1.6 [Reference SimonSim20, Proposition 2.6].

Let $\pi $ be a partial type generically stable over A. Then:

(FS) $\pi $ is finitely satisfiable in every model containing A;

(NF) let $\phi (x;b)\in \pi $ , and take $a\vDash \pi |A$ such that $\vDash \neg \phi (a;b)$ . Then both $\mathrm {tp}(b/Aa)$ and $\mathrm {tp}(a/Ab)$ fork over A.

Fact 1.7 [Reference SimonSim20, Lemma 2.9].

Let $\pi (x)$ be generically stable over A, and let $\pi _0(x)\subseteq \pi (x)$ be a partial ind-definable type, ind-definable over some $A_0\subseteq A$ (i.e., the parameters in the ind-definitions of $\pi $ come from $A_{0}$ ). Then there is $\pi _*(x)\subseteq \pi (x)$ containing $\pi _0(x)$ which is generically stable and ind-defined over some $A_*\subseteq A$ of size $\leq |A_0|+|T|$ .

Proof. We show by induction on $n<\omega $ that there is $\bar a=(a_i:i<n)$ such that $\bar a\vDash \pi ^{(n)}(x)|_A$ and $\bar a^* \vDash \lambda ^{(n)}(x)|_A$ , where $\bar a^* = (a_{n-1},a_{n-2},\ldots ,a_0)$ . For $n=1$ , this is the hypothesis. Assume we know it for n, witnessed by $\bar a=(a_i:i<n)$ . Since $\pi (x)|_A \cup \lambda (x)|_A$ is consistent so is $\pi (x)|_A \cup \lambda (x)$ . Let $\bar b=(b_i:i<\kappa )$ be a long Morley sequence in that partial type. Since we assume our partial types are closed under logical consequence, the fact that $\overline {a} \vDash \pi ^{(n)}|_{A}$ implies that $\pi ^{(n)} \cup \text {tp}(\overline {a}/A)$ is consistent. Thus, composing by an automorphism over A, we may assume that $\bar a\vDash \pi ^{(n)}|_{A\bar b}$ . By generic stability of $\lambda $ , there is $i<\kappa $ such that $b_i\vDash \lambda |_{A\bar a}$ . It follows that $(b_i)^{\frown }\bar a \vDash \pi ^{(n+1)}|_A$ and that ${\overline {a}^{*}}^{\frown } (b_i) \vDash \lambda ^{(n+1)}|_A$ . This finishes the induction.

This being done, we can construct, by Fact 1.2 and compactness, a sequence $\bar d=(d_i:i<\omega )$ which is a Morley sequence of $\pi $ over A such that the sequence in the reverse order is a Morley sequence of $\lambda $ over A. We can further assume that $\bar d\vDash \pi ^{(\omega )}|_{\mathbb {M}}$ . The set of formulas over $\mathbb {M}$ that are true on almost all elements of $\bar d$ contains $\lambda (x)$ , and therefore $\pi (x)\cup \lambda (x)$ is consistent.

Finally, we conclude that $\pi (x) \cup \lambda (x)$ is generically stable over A. Let $\mu (x)$ be the partial type generated by $\pi (x) \cup \lambda (x)$ . It is clear that $\mu (x)$ is ind-definable over A using Fact 1.2 and the fact that $\{(a,\overline {b}) : \overline {b} \in \mathbb {M}^{\omega }, a \vDash \mu (x)|_{A\overline {b}}\}$ is equal to the intersection $\{(a,\overline {b}) : \overline {b} \in \mathbb {M}^{\omega }, a \vDash \pi (x)|_{A\overline {b}}\} \cap \{(a,\overline {b}) : \overline {b} \in \mathbb {M}^{\omega }, a \vDash \lambda (x)|_{A\overline {b}}\}$ . If $\varphi (x;b) \in \mu (x)$ , then there are $\psi _{0}(x;c) \in \pi (x)$ and $\psi _{1}(x;d) \in \lambda (x)$ such that $\psi _{0}(x;c) \wedge \psi _{1}(x;d) \vdash \varphi (x;b)$ . Taking $I = (a_{i} : i < \omega ) \vDash \mu ^{(\omega )}|_{A}$ , since I is Morley over A in both $\pi $ and $\lambda $ , we know that both $\{i : \vDash \psi _{0}(a_{i};c)\}$ and $\{i : \vDash \psi _{1}(a_{i};d)\}$ are cofinite so $\{i : \vDash \varphi (a_{i};b)\}$ is cofinite as well. This shows $\mu $ is generically stable over A.

Proof. By Lemma 1.8, if $\pi (x)$ and $\lambda (x)$ are two generically stable partial types consistent with p, ind-definable over A, then $\pi (x)\cup \lambda (x)$ is consistent and even generically stable over A. Hence, we can define $\pi _p(x)$ as the union of all generically stable partial types consistent with p and ind-definable over A. Then $\pi _p(x)$ is consistent and is the maximal A-invariant generically stable partial type consistent with p. As p itself is generically stable over A, it follows that $\pi _{p}$ extends p.

Proof. Let $\pi _{0}(x)$ be the global partial type defined by closing the set of formulas

$$ \begin{align*}S = p(x) \cup \bigcup \{\neg E(x;c) : c \in \mathbb{M}\} \end{align*} $$

under conjunction and logical consequence. Then $\pi _{0}$ is a consistent partial type by our assumption that $E(x,y)$ is $\bigvee $ -definable and has unboundedly many classes among realizations of p. We have $\pi _{0}$ is ind-definable over A since, writing $E(x,y) = \bigvee \psi _{i}(x,y)$ , we can ind-define $\pi _{0}$ (on the generating formulas) via the schema $(\varphi (x), y=y)_{\varphi (x) \in p}$ and $(\neg \psi _{i}(x,y), y = y)_{i}$ . If $(a_{i} : i < \omega )$ is a sequence with $a_{i} \vDash \pi _{0}|_{Aa_{<i}}$ , then we have $\neg E(a_{i},a_{j})$ for all $i \neq j$ . Therefore, if $c \in \mathbb {M}$ , then c can be E-equivalent to at most one $a_{i}$ . Therefore, if $\chi (x,c) \in S$ , then we have $\vDash \chi (a_{j},c)$ for all but at most one j. Since S generates $\pi _{0}$ , this shows $\pi _{0}$ is a generically stable partial type over A and is therefore contained in the maximal one extending p by Corollary 1.9.

Proof. Note that for any set $B\supseteq A$ , $\eta | B = (\exists y)(\pi (x,y)|_B)$ .

Since $\pi (x,y)$ is A-invariant, $\eta (x)$ is also A-invariant. We first show that $\eta $ is ind-definable using Fact 1.2. Fix a variable $\bar z$ , and let $X(x,y,\bar z)$ be the set of triples $\{(a,b,\bar c) :(a,b) \vDash \pi |A\bar c\}$ . For any tuples a and $\bar c$ , we have $a\vDash \eta |A\bar c$ if and only if there is b such that $(a,b,\bar c)\in X$ . As X is type-definable by Fact 1.2, this whole condition is type-definable. By one more application of Fact 1.2, $\eta $ is ind-definable.

We next show (GS). Assume for a contradiction that for some $\phi (x;c)\in \eta $ , the set $\eta ^{(\omega )}(x_k:k<\omega ) \cup \{\neg \phi (x_k;c) : k<\omega \}$ is consistent. Let $(a_k)_{k<\omega }$ realize it. Note that if we replace $(a_k:k<\omega )$ by a sequence $(a^{\prime }_k:k<\omega )$ which has the same type over A, then we can find $c'\equiv _A c$ such that $\neg \phi (a^{\prime }_k;c')$ holds for all k. By invariance of $\eta $ , we have $\phi (x;c')\in \eta $ , so $(a^{\prime }_k : k < \omega )$ also witnesses a failure of (GS).

Choose $b_0$ such that $(a_{0},b_{0}) \models \pi |_{A}$ , which is possible since $a_0\vDash \eta |A$ . We build by induction on k tuples $(b_k:k<\omega )$ such that $\mathrm {tp}(a_k,b_k/A)=\mathrm {tp}(a_{0},b_{0}/A)$ and $(a_k,b_k)\vDash \pi | Aa_{<k}b_{<k}$ . Assume we have found $b_k$ . As $a_{k+1}\vDash \eta |Aa_{\leq k}$ , there is an automorphism $\sigma $ fixing $Aa_{\leq k}$ such that $\sigma (a_{k+1})\vDash \eta | Aa_{\leq k}b_{\leq k}$ . By the remark above, we may replace the sequence $a_{>k}$ by $\sigma (a_{>k})$ since this does not alter the type of the full sequence $(a_i)_{i<\omega }$ . Hence, we may assume that actually $a_{k+1}\vDash \eta |Aa_{\leq k}b_{\leq k}$ and then we find $b_{k+1}$ as required.

We now have a sequence $(a_kb_k:k<\omega )$ such that $(a_k,b_k)_{k<\omega }\vDash \pi ^{(\omega )}(x_k: k<\omega )$ and c such that $\phi (x;c)\in \pi $ and $\neg \phi (a_k;c)$ holds for all k. Since the condition $(a_k,b_k)_{k<\omega }\vDash \pi ^{(\omega )}(x_k :k<\omega )|_{A}$ is type definable by Fact 1.2, we can apply Ramsey and compactness and assume that the sequence $(a_kb_k:k<\omega )$ is indiscernible over $Ac$ . Using (GS) for the type $\pi $ , we conclude that for every k, $(a_k,b_k)\vDash \pi |Ac$ . But by the definition of $\eta $ , this means that $a_k\vDash \eta |Ac$ . Contradiction.

Fact 3.3 [Reference Kim, Kim and ScowKKS14, Theorem 4.3] [Reference Takeuchi and TsuboiTT12, Theorem 16].

Let denote I _s be the L _{s, ω} -structure (ω ^<ω , ⊴, < _lex , $\wedge$ , (P _α ) _α<ω ) with all symbols being given their intended interpretations and each P _α naming the elements of the tree at level α, and let $I_{0}$ denote its reduct to $L_{0} = \{ \unlhd , \leq _{lex}, \wedge \}$ . Then both $I_{0}$ -indexed indiscernibles (strongly indiscernible trees) and I _s -indexed indiscernibles (s-indiscernible trees) have the modeling property.

Fact 3.5 [Reference ScowSco12, Theorem 3.12].

Suppose I is an infinite, locally finite structure expanding a linear order in the language $L'$ , such that quantifier-free types are isolated by quantifier-free formulas. Then I-indexed indiscernibles have the modeling property if and only if $\mathrm {Age}(I)$ is a Ramsey class.

Proof. Since $\overline {\eta }$ and $\overline {\nu }$ are $\wedge $ -closed and $\mathrm {qftp}_{L_{0,P}}(\overline {\eta }) = \mathrm {qftp}_{L_{0,P}}(\overline {\nu })$ , it is enough to show that the map $\overline {\eta } \mapsto \overline {\nu }$ preserves every predicate of the form $P_{i}$ for $i \leq \omega $ . But this mapping takes $\overline {\eta }_{-}$ to $\overline {\nu }_{-}$ so preserves $P_{i}$ for every $i < \omega $ . The mapping also takes $\overline {\eta }_{+}$ to $\overline {\nu }_{+}$ so preserves $P_{\omega }$ as well.

Proof. Let $(a^{\prime }_{\eta })_{\eta \in \omega ^{\leq \omega }}$ be an s-indiscernible tree locally based on $(a_{\eta })_{\eta \in \omega ^{\leq \omega }}$ .

So now let $\mathrm {EM}_{L_{0,P}}((a^{\prime }_{\eta })_{\eta \in \omega ^{\leq \omega }})$ denote the partial type in the variables $(x_{\eta })_{\eta \in \omega ^{\leq \omega }}$ consisting of the following set of formulas:

$$ \begin{align*}\{\varphi(x_{\overline{\eta}}) : \mathbb{M} \vDash \varphi(a^{\prime}_{\overline{\nu}}) \text{ for all }\overline{\nu} \vDash \mathrm{qftp}_{L_{0,P}}(\overline{\eta})\}. \end{align*} $$

Let $\Gamma $ denote the partial type consisting of $\mathrm {EM}_{L_{0,P}}((a^{\prime }_{\eta })_{\eta \in \omega ^{\leq \omega }})$ , and the collection of formulas asserting that $(x_{\eta })_{\eta \in \omega ^{\leq \omega }}$ is treetop indiscernible. By Claim 3.9, it suffices to show $\Gamma $ is consistent. A finite subset of $\Gamma $ will be contained in

$$ \begin{align*}\mathrm{EM}_{L_{0,P}}((a^{\prime}_{\eta})_{\eta \in \omega^{\leq \omega}})|_{x_{\overline{\xi}}} \cup \left\{ \mathrm{tp}_{\Delta}(x_{\overline{\eta}_{i}}) = \mathrm{tp}_{\Delta}(x_{\overline{\nu}_{i}}) : i < k\right\} \end{align*} $$

for some finite $\Delta $ , a finite tuple $\overline {\xi }$ from $\omega ^{\leq \omega }$ and $\wedge $ -closed tuples $\overline {\eta }_{i},\overline {\nu }_{i}$ with $\overline {\nu }_{i} \vDash \mathrm {qftp}_{L_{0,P}}(\overline {\eta }_{i})$ for all $i < k$ . Let C be a finite $L_{0,P}$ -substructure of $\omega ^{\leq \omega }$ containing $\overline {\xi }$ and $\overline {\eta }_{i}, \overline {\nu }_{i}$ for all $i < k$ and so $C_{-}$ is the $L_{0}$ -substructure of $\omega ^{<\omega }$ consisting of the elements of $C \setminus P(C)$ .

For each $i < k$ , let $q_{i} = \mathrm {qftp}_{L_{0,P}}(\overline {\eta }_{i})$ and define a coloring $c_{i} : q_{i}(\omega ^{\leq \omega }) \to S^{l(\overline {\eta }_{i})}_{\Delta }(\emptyset )$ by

$$ \begin{align*}c_{i}(\overline{\zeta}) = \mathrm{tp}_{\Delta}(a^{\prime}_{\overline{\zeta}}) \end{align*} $$

for all $\overline {\zeta } \in q_{i}(\omega ^{\leq \omega })$ . Note that, since $\Delta $ is finite, we know $S^{l(\overline {\eta }_{i})}_{\Delta }(\emptyset )$ is finite.

Let, for each $i < k$ , $\overline {\eta }_{-,i}$ be the subtuple of $\overline {\eta }_{i}$ consisting of those elements not in $\omega ^{\omega }$ and likewise for $\overline {\nu }_{-,i}$ . Let $q_{-,i} = \mathrm {qftp}_{L_{0}}(\overline {\eta }_{-,i}) = \mathrm {qftp}_{L_{0}}(\overline {\nu }_{-,i})$ . Then we define a coloring $c_{-,i} : q_{-,i}(\omega ^{<\omega }) \to S^{l(\overline {\eta }_{i})}_{\Delta }(\emptyset )$ by setting, for each $\overline {\mu } \in q_{-,i}(\omega ^{<\omega })$ ,

$$ \begin{align*}c_{-,i}(\overline{\mu}) = c_{i}(\overline{\zeta}) = \mathrm{tp}_{\Delta}(a^{\prime}_{\overline{\zeta}}) \end{align*} $$

for any $\overline {\zeta } \in q_{i}(\omega ^{\leq \omega })$ with $\overline {\zeta }_{-} = \overline {\mu }$ . By Lemma 3.7 and the s-indiscernibility of $(a^{\prime }_{\eta })_{\eta \in \omega ^{\leq \omega }}$ , $c_{-,i}$ is well-defined. As $\mathrm {Age}_{L_{0}}(\omega ^{<\omega })$ is a Ramsey class, by Fact 3.3, there is some $C_{-}' \cong C_{-}$ , an $L_{0}$ -substructure of $\omega ^{<\omega }$ , such that $c_{-,i}|_{q_{-,i}(C^{\prime }_{-})}$ is constant for all $i < k$ . Choose any $C' \supseteq C^{\prime }_{-}$ , with $C'$ a substructure of $\omega ^{\leq \omega }$ and $C'$ isomorphic to C as an $L_{0,P}$ -structure. Then, unravelling definitions, we have that $c_{i}|_{q_{i}(C')}$ is constant for all $i < k$ . Letting $\overline {\xi }'$ , $\overline {\eta }^{\prime }_{i}$ and $\overline {\nu }^{\prime }_{i}$ denote the corresponding tuples in $C'$ , we have that $a^{\prime }_{\overline {\xi '}}$ , $(a^{\prime }_{\overline {\eta }_{i}})_{i < k}$ and $(a^{\prime }_{\overline {\nu }_{i}})_{i < k}$ realize the desired finite subset of $\Gamma $ . This concludes the proof.

Proof. Immediate by Lemma 3.8 and Fact 3.5.

Proof. The implication $(2)\implies (1)$ is trivial and $(1) \implies (3)$ is easy, using that $\mathrm {Age}(\omega ^{\leq \omega }) = \mathrm {Age}(\mathcal {T})$ , so we show $(3)\implies (2)$ . Assume (3), and suppose $\mathcal {S}$ is an $L_{0,P}$ -structure with $\mathrm {Age}(\mathcal {S})=\mathbb {K}_{0,P}$ , $(a_{\eta } : \eta \in \mathcal {S})$ is a treetop indiscernible and $\xi \in \mathcal {S}_{-}$ . We must show $(a_{\eta } : \eta \in C(\xi ))$ is order-indiscernible over $a_{\xi }$ . By assumption, $S_{\xi }$ satisfies $\mathrm {Age}(S_{\xi }) \supseteq \mathrm {Age}(\omega ^{\leq \omega })$ . Consequently, for each finite tuple $\overline {\eta }$ from $\omega ^{\leq \omega }$ , there is some $\overline {\nu }$ in $\mathcal {S}_{\xi }$ such that $\mathrm {qftp}_{L_{0,P}}(\overline {\eta }) = \mathrm {qftp}_{L_{0,P}}(\overline {\nu })$ . We define the type $p_{\overline {\eta }}(x_{\overline {\eta }})$ to be $\text {tp}(a_{\overline {\nu }})$ for some (equivalently, all) such $\overline {\nu }$ . Then, by compactness, $\Gamma (x_{\eta } : \eta \in \omega ^{\leq \omega }) = \bigcup _{\overline {\eta }} p_{\overline {\eta }}$ is consistent, where $\overline {\eta }$ ranges over all finite tuples of $\omega ^{\leq \omega }$ . Moreover, letting $(b_{\eta } : \eta \in \omega ^{\leq \omega })$ be a realization, we have that $(b_{\eta } : \eta \in \omega ^{\leq \omega })$ is treetop indiscernible. By assumption, then, $(b_{\eta } : \eta \in \omega ^{\omega })$ is order indiscernible over $b_{\emptyset }$ . By construction, this entails that $(a_{\eta } : \eta \in C(\xi ))$ is order indiscernible over $a_{\xi }$ . As the case of $\xi \in \mathcal {S}_{+}$ is trivial, this completes the proof.

Proof. Suppose $(a_{\eta })_{\eta \in \omega ^{\leq \omega }}$ is treetop indiscernible. We must show that $(a_{\eta })_{\eta \in \omega ^{\omega }}$ is indiscernible over $a_{\emptyset }$ . By compactness, we may stretch the given treetop indiscernible to $(a_{\eta })_{\eta \in \kappa ^{\leq \omega }}$ with $\kappa = |T|^{+}$ . Since T is NIP, by [Reference SimonSim15, Proposition 2.8], there is an end segment $J \subseteq \kappa ^{\omega }$ such that $(a_{\eta })_{\eta \in J}$ is $a_{\emptyset }$ -indiscernible. By treetop indiscernibility, it follows that $(a_{\eta })_{\eta \in \kappa ^{\omega }}$ is $a_{\emptyset }$ -indiscernible as well. Therefore, T is treeless.

Question 3.14. Is Proposition 3.13 true without the assumption that T is NIP? Note that weakened notion of treeless, in which the leaves indexed by $\omega ^{\omega }$ in a treetop indiscernible $(a_{\eta })_{\eta \in \omega ^{\leq \omega }}$ are only required to be an indiscernible sequence (not necessarily indiscernible over $a_{\emptyset }$ ) suffices for many of the observations.

Question 3.15. To check treelessness, does it suffice to consider triples of leaves? More precisely, if whenever $(a_{\eta })_{\eta \in \mathcal {T}}$ is a treetop indiscernible and, for all $\eta _{0} <_{lex} \eta _{1} <_{lex} \eta _{2}$ and $\nu _{0} <_{lex} \nu _{1} <_{lex} \nu _{2}$ from $\mathcal {T}_{-}$ , we have $(a_{\eta _{0}},a_{\eta _{1}}, a_{\eta _{2}}) \equiv _{a_{\emptyset }} (a_{\nu _{0}}, a_{\nu _{1}}, a_{\nu _{2}})$ , does it follow that T is treeless?

Proof. Suppose $T'$ is interpretable in T and E is a T-definable equivalence relation such that if $M \vDash T$ , then $M^{n}/E$ is the domain of a model of $T'$ whose relations are definable in T. Let $\mathbb {M}' = \mathbb {M}^{n}/E$ and let $\pi : \mathbb {M}^{n} \to \mathbb {M}'$ denote the interpretation map. Suppose $(a_{\eta })_{\eta \in \omega ^{\leq \omega }}$ is a treetop indiscernible in $\mathbb {M}'$ . Then for each $\eta \in \omega ^{\leq \omega }$ , we can choose some $\tilde {a}_{\eta } \in \pi ^{-1}(a_{\eta })$ . We can then take $(b_{\eta })_{\eta \in \omega ^{\leq \omega }}$ which is treetop indiscernible and locally based on $(\tilde {a}_{\eta })_{\eta \in \omega ^{\leq \omega }}$ in $\mathbb {M}$ . As T is treeless, $(b_{\eta })_{\eta \in \omega ^{\omega }}$ is an indiscernible sequence over $b_{\emptyset }$ . In particular, $(\pi (b_{\eta }))_{\eta \in \omega ^{\omega }}$ is an indiscernible sequence over $\pi (b_{\emptyset })$ . But since $(a_{\eta })_{\eta \in \omega ^{\leq \omega }}$ was taken to be treetop indiscernible in $\mathbb {M}'$ , we have, by local basedness, that $(a_{\eta })_{\eta \in \omega ^{\leq \omega }} \equiv (\pi (b_{\eta }))_{\eta \in \omega ^{\leq \omega }}$ , hence $(a_{\eta })_{\eta \in \omega ^{\omega }}$ is an indiscernible sequence over $a_{\emptyset }$ , which shows $T'$ is treeless.

Proof. We prove the contrapositive. Suppose T has $2$ -IP witnessed by the formula $\varphi (x;y,z)$ . Then, by compactness, there is a sequence $(b_{\eta }, c_{\eta } : \eta \in \omega ^{\omega })$ such that, for all $X \subseteq \omega ^{\omega } \times \omega ^{\omega } $ , there is some $a_{X}$ such that

$$ \begin{align*}\vDash \varphi(a_{X};b_{\eta}, c_{\nu}) \iff (\eta, \nu) \in X. \end{align*} $$

Now, for each $\eta \in \omega ^{\omega }$ , let

$$ \begin{align*}X_{\eta} = \{(\nu, \xi) \in \omega^{\omega} \times \omega^{\omega} : \eta <_{lex} \nu <_{lex} \xi \text{ and } \eta \wedge \nu \vartriangleleft \nu \wedge \xi \}. \end{align*} $$

Choose, for each $\eta \in \omega ^{\omega }$ some $a_{\eta }$ such that

$$ \begin{align*}\vDash \varphi(a_{\eta}; b_{\nu}, c_{\xi}) \iff (\nu, \xi) \in X_{\eta}. \end{align*} $$

Choose a sequence of same-length tuples $(d_{\eta })_{\eta \in \omega ^{<\omega }}$ arbitrarily and set $d_{\eta } = (a_{\eta }, b_{\eta }, c_{\eta })$ for each ${\eta \in \omega ^{\omega }}$ . Let $(d^{\prime }_{\eta })_{\eta \in \omega ^{\leq \omega }}$ be a treetop indiscernible locally based on $(d_{\eta })_{\eta \in \omega ^{\leq \omega }}$ , and write $d^{\prime }_{\eta } = (a^{\prime }_{\eta }, b^{\prime }_{\eta }, c^{\prime }_{\eta })$ for each $\eta \in \omega ^{\omega }$ . Note that we still have

$$ \begin{align*}\vDash \varphi(a^{\prime}_{\eta}, b^{\prime}_{\nu},c^{\prime}_{\xi}) \iff \eta <_{lex} \nu <_{lex} \xi \text{ and } \eta \wedge \xi \vartriangleleft \nu \wedge \xi. \end{align*} $$

Choosing $\eta _{0} <_{lex} \eta _{1} <_{lex} \eta _{2} <_{lex} \eta _{3}$ in $\omega ^{\omega }$ with $\eta _{0} \wedge \eta _{1} \vartriangleright \eta _{1} \wedge \eta _{3}$ and $\eta _{0} \wedge \eta _{2} \vartriangleleft \eta _{2} \wedge \eta _{3}$ , we have $\vDash \neg \varphi (a^{\prime }_{\eta _{0}}, b^{\prime }_{\eta _{1}}, c^{\prime }_{\eta _{3}})$ and $\vDash \varphi (a^{\prime }_{\eta _{0}}, b^{\prime }_{\eta _{2}}, c^{\prime }_{\eta _{3}})$ , so $(d^{\prime }_{\eta })_{\eta \in \omega ^{\omega }}$ is not order-indiscernible.

Proof. Let $\phi (x,y,z)$ be a formula in L (without parameters). Let $S^\pi _\phi $ be the collection of formulas $\psi (y,z) \in L(A)$ such that for all d, $\phi (x,c,d) \in \pi $ if and only if $\psi (c,d)$ holds for some $\psi \in S_\phi $ . Note that (*) for any $b\equiv _A c$ , $S^{\pi (x,b)}_\phi = S^\pi _\phi $ , so we can discard the $\pi $ in the notation and write $S_\phi $ . Let $(\pi \rtimes \lambda ) (x,y)$ be the closure under finite conjunctions and logical consequences of $\lambda (y) \cup \{\psi (y,d) \to \phi (x,y,d) : \phi (x,y,z)\in L,\psi (y,z) \in S_{\phi }\}$ . Note that $\pi \rtimes \lambda $ is ind-definable over A as $\lambda (y)$ is and the second part is ind-definable by the defining scheme $(\psi (y;z) \to \phi (x,y;z);z=z)$ , where $\psi \in S_\phi $ . It clearly satisfies the requirement by (*) above.

Uniqueness follows by the fact a global partial type is determined by the realizations of its restrictions to small sets. In the case when $\pi $ is ind-definable over A, note that $\pi (x,b) = \pi $ for any $b\equiv _A c$ , so that uniqueness implies that $\pi \rtimes \lambda = \pi \otimes \lambda $ .

Proof. $\pi \rtimes \lambda $ is ind-definable over A by Lemma 4.3.

We show that $\pi \rtimes \lambda $ is generically stable over A. Let $(a_i,b_i:i<\omega )$ be a Morley sequence in $\pi \rtimes \lambda $ over A. Assume for a contradiction that there is d and a formula $\phi (x,y;z)\in L(A)$ so that $\neg \phi (x,y;d)\in \pi \rtimes \lambda $ and $\bigwedge _{i<\omega } \phi (a_i,b_i;d)$ holds. We may assume that $(a_i,b_i:i<\omega )$ is $Ad$ -indiscernible.

As $b \in \operatorname {acl}(A)$ and $(b_i : i<\omega )$ is A-indiscernible in the type of b over A, there is some $b'$ such that $b_i=b'$ for all $i<\omega $ . Thus, we have that $b' \vDash \lambda |_A$ (trivially) and $a_i \vDash \pi (x,b')|_{Aa_{<i}b'}$ for all $i<\omega $ . Additionally, $(a_i : i<\omega )$ is indiscernible over $Adb'$ so by generic stability $a_i \vDash \pi (x,b')|_{Aa_{<i}b'd}$ , contradiction.

Proof. The type $\pi \rtimes \lambda $ is ind-definable over A by Lemma 4.3.

We show that $\pi \rtimes \lambda $ is generically stable over A. Let $(a_i,b_i:i<\omega )$ be a Morley sequence in $\pi \rtimes \lambda $ over A. Assume for a contradiction that there is d and a formula $\phi (x,y;z)\in L(A)$ so that $\neg \phi (x,y;d)\in \pi \rtimes \lambda $ and $\bigwedge _{i<\omega } \phi (a_i,b_i;d)$ holds. We may assume that $(a_i,b_i:i<\omega )$ is $Ad$ -indiscernible. By generic stability of $\lambda $ , it follows that $(b_i:i<\omega )$ is a Morley sequence of $\lambda $ over $Ad$ .

We extend this sequence to a tree $(c_{\eta } : \eta \in \omega ^{<\omega })$ so that:

• ○ For $\eta \in \omega ^{<\omega }$ ,

  $$ \begin{align*}c_{\eta} = (\overline{a}_{\eta}, b_{\eta}) = ((a_{\eta,i} : i < \omega), b_{\eta}), \end{align*} $$
  where $(a_{\eta ,i} : i < \omega )$ is a Morley sequence in $\pi (x;b_{\eta })$ over $Ab_{\eta }$ .

• ○ For every $\eta \in \omega ^{\omega }$ , the sequence $(a_{\eta | i, \eta (i)}, b_{\eta | i} : i < \omega )$ has the same type as $(a_{i}, b_{i} : i < \omega )$ over A.

To build the tree, we start by taking a Morley sequence $(a^{\prime }_{i} : i < \omega )$ in $\pi (x;b_{0})|_{Ab_{0}}$ with $a^{\prime }_{0} = a_{0}$ , and then we set $c_{\emptyset } = ((a^{\prime }_{i} : i < \omega ), b_{0})$ . Assume we have constructed $(c_{\eta })_{\eta \in \omega ^{< k}}$ such that, for all $\eta \in \omega ^{k}$ ,

$$ \begin{align*}(a_{\eta | i, \eta(i)}, b_{\eta | i} : i < k) \equiv_{A} (a_{i}b_{i} : i < k). \end{align*} $$

Fix $\eta \in \omega ^{k}$ . Then we choose $a'$ and $b'$ so that

$$ \begin{align*}(a_{\eta | i, \eta(i)}, b_{\eta | i} : i < k)a'b' \equiv_{A} (a_{i}b_{i} : i < k)a_{k}b_{k}. \end{align*} $$

Since $a_{k} \vDash \pi (x;b_{k})|_{Aa_{<k}b_{\leq k}}$ , we know $a'$ satisfies $\pi (x;b')$ restricted to $A(a_{\eta | i, \eta (i)}, b_{\eta | i} : i < k)b'$ and therefore

$$ \begin{align*}\pi(x;b') \cup \text{tp}(a'/A(a_{\eta | i, \eta(i)}, b_{\eta | i} : i < k)b') \end{align*} $$

is consistent. We then choose $\overline {a}' = (a^{\prime }_{i} : i< \omega )$ to be a Morley sequence over $A(a_{\eta | i, \eta (i)}, b_{\eta | i} : i < k)b'$ in this type with $a^{\prime }_{0} = a'$ and define $c_{\eta } = (\overline {a}_{\eta }, b_{\eta })$ by setting $\overline {a}_{\eta } = \overline {a}'$ and $b_{\eta } = b'$ . This defines $(c_{\eta } : \eta \in \omega ^{< k+1})$ and thus, by induction, all of $(c_{\eta })_{\eta \in \omega ^{<\omega }}$ .

For each branch $\eta \in \omega ^{\omega }$ , we can find $d_\eta $ such that $d_{\eta }(a_{\eta |n, \eta (n)},b_{\eta |n})_{n < \omega } \equiv _A d (a_{n},b_{n})_{n < \omega }$ and hence such that $\bigwedge _{n<\omega } \phi (a_{\eta |n, \eta (n)},b_{\eta |n};d_\eta )$ holds. Define a tree $(e_{\eta } : \eta \in \omega ^{\leq \omega })$ by setting $e_{\eta } = b_{\eta }$ for $\eta \in \omega ^{<\omega }$ and $e_{\eta } = d_{\eta }$ for $\eta \in \omega ^{\omega }$ . Let $(e^{\prime }_\eta :\eta \in \omega ^{\omega }) = (b^{\prime }_{\eta } :\eta \in \omega ^{<\omega })^{\frown } (d^{\prime }_{\eta } : \eta \in \omega ^{\omega })$ be treetop indiscernible over A, locally based on $(e_{\eta } : \eta \in \omega ^{\leq \omega })$ . By compactness, we can stretch our treetop indiscernible to $(e^{\prime }_{\eta } : \eta \in \kappa ^{\leq \omega })$ for $\kappa = (|T|+|A|)^{+}$ . By treelessness, then, $(d^{\prime }_{\eta }:\eta \in \kappa ^{\omega })$ is order-indiscernible over $Ab^{\prime }_{\emptyset }$ .

By induction on n, we will build a path $\eta _{*}|(n+1) \in \kappa ^{n+1}$ and sequences $\overline {a}_{n} = (a_{n,i} : i < \kappa )$ for each $n < \omega $ such that

• ○ For all $i \leq n$ , $\overline {a}_{i}$ realizes $\pi ^{(\kappa )}(x;b^{\prime }_{\emptyset })$ over $Ab^{\prime }_{\emptyset }$ .

• ○ $(a_{i,\eta _{*}(i)} : i \leq n)$ realizes $\pi ^{(n+1)}(x;b^{\prime }_{\emptyset })$ over $Ab^{\prime }_{\emptyset }$ .

• ○ For all $\eta \in C(\eta _{*}|n)$ , $\vDash \phi (a_{n,\eta _{*}(n)}, b^{\prime }_{\emptyset }; d^{\prime }_{\eta })$ , where we recall that $C(\nu ) = \{\eta \in \kappa ^{\omega } : \nu \unlhd \eta \}$ .

As $(e^{\prime }_{\eta }: \eta \in \kappa ^{\leq \omega })$ is locally based on $(e_{\eta } : \eta \in \omega ^{\leq \omega })$ , we know

$$\begin{align*}(\exists \overline{x})\left[ \pi^{(\kappa)}(\overline{x};b^{\prime}_{\emptyset})|_{Ab^{\prime}_{\emptyset}} \wedge \bigwedge_{i < \kappa} \bigwedge_{\eta \in C(\langle i \rangle)} \phi(x_{i},b^{\prime}_{\emptyset};d_{\eta})\right] \end{align*}$$

since this follows from

$$\begin{align*}(\exists \overline{x})\left[ \pi^{(\omega)}(\overline{x};b_{\nu})|_{Ab_{\nu}} \wedge \bigwedge_{i < \omega} \bigwedge_{\eta \in C(\nu^{\frown}\langle i\rangle)} \phi(x_{i},b_{\nu};d_{\eta})\right], \end{align*}$$

and this was witnessed by $(a_{\nu ,i} : i < \omega )$ . Therefore, we can let $\overline {a}_{0} = (a_{0,i} : i < \kappa )$ be a Morley sequence in $\pi (x;b^{\prime }_{\emptyset })$ such that $\vDash \bigwedge _{i < \omega } \bigwedge _{\eta \in C(\langle i\rangle )} \phi (a_{0,i},b^{\prime }_{\emptyset },d_{\eta })$ . We set $\eta _{*}(0) = 0$ .

Assume we have constructed $\eta _{*}|k$ for $k>0$ . Since $(d^{\prime }_{\eta } : \eta \in \kappa ^{\omega })$ is an indiscernible sequence over $Ab^{\prime }_{\emptyset }$ , we know

$$ \begin{align*}\langle (d^{\prime}_{\eta})_{\eta \in C(\langle i \rangle)}: i < \kappa \rangle \equiv_{Ab^{\prime}_{\emptyset}} \langle (d^{\prime}_{\eta})_{\eta \in C((\eta_{*}|k)^{\frown} \langle i\rangle)}: i < \kappa \rangle. \end{align*} $$

Choose $\overline {a}_{k}$ such that

$$ \begin{align*}\overline{a}_{0}\langle (d^{\prime}_{\eta})_{\eta \in C(\langle i\rangle)}: i < \kappa \rangle \equiv_{Ab_{\emptyset}'} \overline{a}_{k} \langle (d^{\prime}_{\eta})_{\eta \in C((\eta_{*}|k)^{\frown}\langle i\rangle)}: i < \kappa \rangle. \end{align*} $$

Then $\overline {a}_{k}$ is a Morley sequence in $\pi (x;b^{\prime }_{\emptyset })$ over $Ab^{\prime }_{\emptyset }$ and we have

$$ \begin{align*}\vDash \phi(a_{k,i}, b^{\prime}_{\emptyset}, d^{\prime}_{\eta}) \end{align*} $$

for all $\eta \in \kappa ^{\omega }$ with $\eta _{*}|k \unlhd \eta $ and $\eta (k) = i$ , by the choice of $\overline {a}_{k}$ and the properties of $\overline {a}_{0}$ . By generic stability and the choice of $\kappa $ , there is some $i_{*} < \kappa $ such that

$$ \begin{align*}a_{k,i_{*}} \vDash \pi(x;b_{\emptyset})|_{A(a_{i,\eta_{*}(i)})_{i < k}}. \end{align*} $$

Then we set $\eta _{*}(k) = i_{*}$ .

We have constructed a path $\eta _* \in \kappa ^{\omega }$ so that $(a_{n, \eta _*(n)})_{n < \omega }$ realizes $\pi ^{(\omega )}(x,b^{\prime }_{\emptyset })$ over $Ab^{\prime }_{\emptyset }$ . Extracting, we may assume that this sequence is indiscernible over $Ab^{\prime }_{\emptyset }d^{\prime }_{\eta _*}$ . Now, $b^{\prime }_{\emptyset }\vDash \lambda |_{Ad^{\prime }_{\eta _*}}$ and since $\pi (x,b^{\prime }_{\emptyset })$ is generically stable over $Ab^{\prime }_{\emptyset }$ , also $(a_{n,\eta _*(n)}:n<\omega )$ is a Morley sequence of $\pi (x,b^{\prime }_{\emptyset })$ over $Ab^{\prime }_{\emptyset }d^{\prime }_{\eta _*}$ . Hence, $(a_{0, \eta _{*}(0)},b^{\prime }_{\emptyset })\vDash \pi \rtimes \lambda |_{Ad^{\prime }_{\eta _*}}$ . Contradiction.

Proof. Pick any $c \vDash \lambda |_{A}$ , and let $\lambda ' = \lambda \cup \text {tp}(c/A)$ . Then by the ‘it follows’ part of Lemma 4.3, $\pi \rtimes \lambda ' = \pi \otimes \lambda '$ . Thus, $(\pi \otimes \lambda ')$ is generically stable by Proposition 4.6.

Suppose, towards contradiction, that $(\pi \otimes \lambda )$ is not generically stable over A. Then there is a sequence $I = (a_{i},b_{i} : i < \omega ) \vDash (\pi \otimes \lambda )^{(\omega )}|_{A}$ and some $\varphi (x,y;d) \in (\pi \otimes \lambda )$ such that $\bigwedge _{i < \omega } \neg \varphi (a_{i},b_{i};d)$ . After extracting, we may assume that I is $Ad$ -indiscernible. Then for $\lambda ' = \lambda \cup \text {tp}(b_{0}/A)$ , we have $I \vDash (\pi \otimes \lambda ')^{(\omega )}|_{A}$ and $\varphi (x,y;d) \in \pi \otimes \lambda '$ , contradicting the generic stability of $(\pi \otimes \lambda ')$ .

Proof. Symmetry follows by Corollary 4.7 and Proposition 2.4. To see base monotonicity, assume . We want to show . If not, then $b \not \vDash \pi (x)|_{Aac}$ , where $\pi (x)$ is the maximal global type extending $\text {tp}(b/Ac)$ which is generically stable over $Ac$ . Let $\lambda (y) = \text {tp}(c/A)$ . Since $\lambda (y)$ is generically stable over A and $\pi (x)$ is generically stable over $Ac$ , Proposition 4.6 implies that $(\pi \rtimes \lambda ) (x,y)$ is generically stable over A. Since $(b,c) \vDash \pi \rtimes \lambda |_{Aa}$ if and only if $c \vDash \lambda |_{Aa}$ and $b \vDash \pi (x,c) = \pi (x)|_{Aac}$ , we know that $(b,c) \not \vDash \pi \rtimes \lambda |_{Aa}$ . However, $\pi \rtimes \lambda $ is consistent with $\text {tp}(bc/A)$ , so this contradicts our assumption that .

Proof. By [Reference AdlerAdl09, Remark 5.5], a theory is rosy if and only if there is a strict independence relation: that is an $\mathrm {Aut}(\mathbb {M})$ -invariant ternary relation on small subsets of $\mathbb {M}$ satisfying the properties listed in Theorem 2.2, plus symmetry, base monotonicity and full existence. Full existence is easily seen to be a consequence of extension and existence so follows from Theorem 2.2 as well. Symmetry and base monotonicity follow from treelessness by Corollary 4.8.

Proof. First, assume . By algebraicity and normality (see Theorem 2.2), . Now, continue as in the proof of Corollary 4.8, using Proposition 4.5 instead of Proposition 4.6 to get . Finally, follows from monotonicity.

For the other direction, assume . Since by algebraicity and left existence, we may apply left transitivity and monotonicity to get .

Fact 6.3. Suppose T is an NSOP $_{1}$ theory. We have the following:

1. 1. Symmetry: if $M \vDash T$ , if and only if . [Reference Kaplan and RamseyKR17, Theorem 5.16]

2. 2. T is simple if and only if satisfies base monotonicity –that is, whenever $M \preceq N \vDash T$ , if , then . [Reference Kaplan and RamseyKR17, Proposition 8.8]

3. 3. Witnessing: If $M \vDash T$ and $I = \langle a_{i} : i < \omega \rangle $ is an -Morley sequence over M with $a_{0} = a$ , then if and only if there is $I' \equiv _{Ma} I$ such that $I'$ is $Mb$ -indiscernible. [Reference Kaplan and RamseyKR21, Theorem 5.1]

4. 4. Lifting lemma: If $M \preceq N \vDash T$ and , then there is an N-indiscernible sequence $I = \langle a_{i} : i < \omega \rangle $ with $a_{0} = a$ that is both -Morley over M and -Morley over N. [Reference Kaplan and RamseyKR21, Proposition 3.3]

Proof. We argue in the NSOP $_1$ case; the proof in the simple case is the same, except that we can drop the assumption that M is a model (in that case Kim-independence is just the usual forking-independence). Let M be a model, and consider a tuple b. Let $p=\mathrm {tp}(b/M)$ . Consider the partial type

$$\begin{align*}\pi(x) = p(x) \cup \{\neg \phi(x;c):c\in \mathbb{M}, \phi(x;c)\cup p(x)\text{ Kim-divides over }M\}.\end{align*}$$

This partial type is M-invariant. To see that it is ind-definable over M, by Fact 1.2, we have to argue that the set

$$ \begin{align*}X = \{(a,\overline{b}) : \overline{b} \in \mathbb{M}^{\omega}, a \vDash \pi|_{M\overline{b}}\} \end{align*} $$

is type-definable over M. Fix $I = (a_{i})_{i < \omega }$ , any coheir sequence over M in p. Let $q(x_{0},x_{1}, \ldots ) = \text {tp}((a_{i})_{i < \omega }/M)$ . Notice that, if $(a,\overline {b}) \in X$ , then, since $a \equiv _{M} a_{0}$ , there is $I' = (a^{\prime }_{i})_{i < \omega }$ with $I' \equiv _{M} I$ and $a^{\prime }_{0} = a$ . By symmetry and the definition of $\pi $ , and hence there is $I" \equiv _{Ma} I'$ which is $M\overline {b}$ -indiscernible. This shows that if $(a,\overline {b}) \in X$ , then there is $I" \vDash q$ which starts with a and which is $M\overline {b}$ indiscernible. On the other hand, if there is some $J \vDash q$ which starts with a and which is $M\overline {b}$ -indiscernible, then by symmetry and Kim’s lemma, so $(a,\overline {b}) \in X$ . This shows that $(a,\overline {b}) \in X$ if and only if

$$ \begin{align*}(\exists \overline{x}) \left[ \overline{x} \vDash q \wedge x_{0} = a \wedge \overline{x} \text{ is }M\overline{b}\text{-indiscernible}\right], \end{align*} $$

which shows X is type-definable over M.

Additionally, $\pi $ is generically stable: If $\langle a_{i} : i < \omega \rangle $ is a sequence with $a_{i} \vDash \pi |_{M a_{<i}}$ for all $i < \omega $ , then we have . Suppose $\neg \varphi (x;b) \in \pi $ , so $\varphi (x;b)$ Kim-divides over M and we must show that $\vDash \neg \varphi (a_{i};b)$ for all but finitely many i. If not, then, after throwing away a co-infinite set, we may assume $\vDash \varphi (a_{i};b)$ for all $i < \omega $ . However, by symmetry, $\varphi (a_{0};y) \cup \mathrm {tp}_{y}(b/M)$ Kim-divides over M and, thus, by [Reference Kaplan and RamseyKR21, Remark 5.3], $\{\varphi (a_{i};y) : i < \omega \} \cup \text {tp}_{y}(b/M)$ is k-inconsistent for some k, a contradiction. This establishes that $\pi $ is generically stable, so implies (by symmetry and the definition of $\pi $ ).

For the converse, assume that and for a contradiction that . Set $p(x,y)=\mathrm {tp}(a,b/M)$ . Let $\pi (y)$ be generically stable over M, consistent with $\mathrm {tp}(b/M)$ but not with $\mathrm {tp}(b/Ma)$ . Fix some $\psi (y;a) \in \mathrm {tp}(b/Ma)$ such that $\neg \psi (y;a) \in \pi $ . Take $(b_i:i<\omega )$ a -Morley sequence in $\mathrm {tp}(b/M)$ . Then by the first part of the proof, it is a -Morley sequence. As , $\bigwedge p(x,b_i)$ is consistent. Let $a'$ be a realization. We have $a' \models \{\psi (b_{i};x) : i < \omega \}$ and, since $a' \equiv _{M} a$ , we have, by the M-invariance of $\pi $ , that $\neg \psi (y;a') \in \pi $ as well. However, by the generic stability of $\pi $ , we know that $\{ i :\models \psi (b_{i};a')\}$ is finite, a contradiction.

Proof. As T is NSOP $_{1}$ , we have, by Proposition 6.4, that over models. By Corollary 4.8, the treelessness of T implies satisfies base monotonicity.

Question 6.6. If T is NSOP $_1$ , is symmetric over an extension base or even over an arbitrary base (see also Question 2.6)?

Proof. We may write $a_{i} = (a_{i,0},\ldots , a_{i,m-1})$ and likewise for $b_{i}$ , for all $i < \omega $ . If $c \in C$ , then, since I and J are C-indiscernible sequences starting with $a_{0} = b_{0}$ , we have

$$ \begin{align*}a_{i,j}c \equiv a_{0,j}c \equiv b_{0,j}c \equiv b_{i,j}c, \end{align*} $$

for all $i < \omega $ and $j < m$ . Since $I \equiv J$ , it follows that any pair of elements selected from $IC$ will have the same type as the corresponding pair from $JC$ , and thus, by binarity, $IC \equiv JC$ . This shows $I \equiv _{C} J$ .

Proof. By Fact 6.3(2), it suffices to show that satisfies base monotonicity. So fix $M \prec N \vDash T$ , and assume . We must show . By symmetry (Fact 6.3(1)), we have and it suffices to establish .

As , there is a sequence $I = \langle a_{i} : i < \omega \rangle $ with $a_{0} = a$ which is simultaneously -Morley over M and over N, by Fact 6.3(4). Because , there is $J \equiv _{Ma} I$ such that J is $Nb$ -indiscernible. By Lemma 6.7, since $I \equiv _{M} J$ and I and J are both N-indiscernible, starting with a, we have $I \equiv _{N} J$ , from which it follows that J is -Morley over N as well. By Fact 6.3(3), this shows , completing the proof.