Fact 2.1 [Reference Tent and Ziegler24].

The dimension has the following properties.

• • Finite character. $\dim (\bar {a}/Y)=\min \{\dim (\bar {a}/B):B\subseteq _{\text {fin}} Y\}$ .

• • Additivity. $\dim (\bar {a}\bar {b}/Z) = \dim (\bar {a}/ Z) + \dim (\bar {b}/\bar {a} Z)$ .

• • Monotonicity. $\dim (\bar {a}/Y) \geq \dim (\bar {a}/Z)$ for $Y \subseteq Z$ .

• • Definability. For each formula $\varphi (\bar {x},\bar {y})$ and $k\leq |\bar {x}|$ , the set

  $$ \begin{align*} \{\bar{b}\in M^{|\bar{y}|}: \dim (\varphi(M, \bar{b}))=k\} \end{align*} $$
  is definable by a formula $d_{\varphi ,k}(\bar {y})$ .

• • $\bigvee $ -Definability. If $\dim (\bar {a}/\bar {b})\leq n$ , then there is a formula $\psi (\bar {x},\bar {y})$ such that $\psi (\bar {x},\bar {b})\in \operatorname {\mathrm {qftp}}(\bar {a}/\bar {b})$ and if $M\models \psi (\bar {a}^{\prime },\bar {b}^{\prime })$ then $\dim (\bar {a}^{\prime }/\bar {b}^{\prime })\leq n$ , for every $\bar {a}^{\prime },\bar {b}^{\prime }$ .

Fact 2.5. [Reference Tent and Ziegler24] $\dim $ -independence has the following properties.

1. 1. Symmetry. if and only if .

2. 2. Transitivity. if and only if and .

3. 3. $\operatorname {\mathrm {acl}}$ -Preservation. if and only if .

4. 4. $\dim $ -Morley sequences. Any non-constant order indiscernible sequence ${\{a_{i}:\ i\in I\}}$ over X is a $\dim $ -Morley sequence, i.e., for any two disjoint subsequences $J_{1}$ and $J_{2}$ of I we have .

5. 5. Strong finite character. If then there exists a formula $\varphi (\bar {z},\bar {x},\bar {y})\in \operatorname {\mathrm {qftp}}_{\mathcal {L}}(\bar {c},\bar {a},\bar {b})$ which $\dim $ -forks, i.e., for each model M and tuples $\bar {a}^{\prime }$ , $\bar {b}^{\prime }$ and $\bar {c}^{\prime }\in M$ if $M\models \varphi (\bar {c}^{\prime },\bar {a}^{\prime },\bar {b}^{\prime })$ then .

Convention 2.7. The structure M is called a free amalgam of $M_{1}$ and $M_{2}$ over $M_{0}$ and denoted by $M_{1}\oplus _{M_{0}} M_{2}$ . Note that M is not unique up to isomorphism.

Observation 2.8. Let $\Sigma (\bar {x})$ be a partial type over X which has a solution with no intersection with $\operatorname {\mathrm {acl}}(X)$ . Then, for every small set $Y\supseteq X$ , there is a solution $\bar {d}^{\prime }$ for $\Sigma $ in M such that $\bar {d}^{\prime }\cap Y=\emptyset $ . Moreover, $\bar {d}^{\prime }$ can be chosen such that $X {\bar {d}^{\prime }}$ and Y are free over X.

Proof By the hypothesis, there is a solution in $M\oplus _{\operatorname {\mathrm {acl}}(X)}M$ which has no intersection with $\operatorname {\mathrm {acl}}(X)$ . Since M is $\lambda $ -saturated one can find a solution $\bar {d}^{\prime }$ in M such that $X {\bar {d}^{\prime }}$ and Y are free over X.

Fact 2.15 [Reference Pourmahdian23, Notation 3.14].

1. 1. $\operatorname {\mathrm {cl}}_{M}(X)$ is the intersection of all closed subsets of $ M$ that contain X.

2. 2. $\operatorname {\mathrm {cl}}_M(A)=\bigcup \{B\subseteq M: A\leqslant _i B\}$ .

3. 3. $\operatorname {\mathrm {cl}}_{M}(X)=\bigcup _{A\subseteq X} \operatorname {\mathrm {cl}}_{M}(A)$ .

Proof By Theorem 3.6, $\mathbb {M}$ is $\aleph _{0}$ -saturated as a model of T. Now we just prove that the third item in Definition 3.8 holds. Consider $\bar {a},\bar {b}$ , $\varphi $ , $\Phi _{0}$ and $\mathcal {C}_{A,B}$ as in the hypothesis of item $3$ . Let $\bar {a}^{\prime }$ in $\mathbb {M}$ be such that $\mathbb {M}\models D_{\varphi ^{\ast },k}(\bar {a}^{\prime })\wedge d_{\varphi {\ast },k}(\bar {a}^{\prime })$ . So there is $\bar {b}^{\prime }\in \mathbb {M}^{|\bar {b}|}$ with $\mathbb {M}\models \varphi (\bar {b}^{\prime },\bar {a}^{\prime })$ , $\bar {b}^{\prime }\cap \operatorname {\mathrm {acl}}(\bar {a}^{\prime })=\emptyset $ , and $\dim (\bar {b}^{\prime }/\bar {a}^{\prime })=k$ . Since $ \operatorname {\mathrm {cl}}(\bar {a}^{\prime })$ is finite, by $\omega $ -saturation of $\mathbb {M}$ and Observation 2.8 we may assume that $\langle \operatorname {\mathrm {cl}}(\bar {a}^{\prime })\rangle \cap \bar {b}^{\prime }=\emptyset $ and $ \bar {b}' $ and $ \langle \operatorname {\mathrm {cl}}(\bar {a}') \rangle $ are free over $ \bar {a} '$ . Consider the $ \mathcal {L} $ -structure generated by $ \operatorname {\mathrm {cl}}(\bar {a}')\bar {b}' $ in $ \mathbb {M} $ and make it into an $ \mathcal {L}_p $ -structure $ N $ by coloring $ \operatorname {\mathrm {cl}}(\bar {a}')\bar {b}' $ with the same colors as $\operatorname {\mathrm {cl}}(\bar {a}) \bar {b} $ and leaving the rest of the elements non-colored. It is clear that $\langle \operatorname {\mathrm {cl}}(\bar {a}')\rangle \leqslant \mathbb {M}$ . We claim that also $\langle \operatorname {\mathrm {cl}}(\bar {a}')\rangle \leqslant {N}$ . By this claim, since $\mathbb {M}$ is $\aleph _{0}$ -rich, there is a strong embedding $f:N\to \mathbb {M}$ which fixes $\langle \operatorname {\mathrm {cl}}(\bar {a}^{\prime })\rangle $ pointwise. Let $f(\bar {b}^{\prime })=\bar {e}$ . Then

$$ \begin{align*} \mathbb{M}\models \big{(}\varphi(\bar{a}^{\prime},\bar{e})\wedge \bigwedge _{p(b_i)}p(e_{i})\wedge \bigwedge_{\neg p(b_i)}\neg p(e_i)\big{)}. \end{align*} $$

Furthermore we prove that

$$ \begin{align*} \mathbb{M}\models \neg\exists\bar{z} \psi_{A,B,C}(\bar{a}^{\prime},\bar{e};\bar{z}), \end{align*} $$

for every $C\in \Phi _{0}$ . Otherwise suppose there exists $\bar {c}^{\prime }$ in $\mathbb {M}$ such that

$$ \begin{align*} \mathbb{M}\models \psi_{A,B,C}(\bar{a}^{\prime},\bar{e};\bar{c}^{\prime}), \end{align*} $$

for some $C\in \Phi _{0}$ . As $\bar {c}^{\prime }$ is disjoint from $\bar {a}^{\prime }$ and $\bar {e}$ , it follows that $\bar {c}^{\prime }\subseteq \operatorname {\mathrm {cl}}(\bar {a}^{\prime }\bar {e})\setminus \bar {a}^{\prime }\bar {e}$ . Let $A^{\prime }$ and $B^{\prime }$ be the union of elements of tuples $\bar {a}^{\prime }$ and $\bar {a}^{\prime }\bar {e}$ , respectively. Since $ \bar {b}' $ and $ \langle \operatorname {\mathrm {cl}}(\bar {a}') \rangle $ are free over $ \bar {a} '$ it follows that $\operatorname {\mathrm {cl}}(B^{\prime })=\operatorname {\mathrm {cl}}(A^{\prime })\oplus _{A^{\prime }}B^{\prime }$ . As $\psi _{A,B,C}$ is an intrinsic formula, we have that $\bar {c}^{\prime }\subseteq \operatorname {\mathrm {cl}}(B^{\prime })$ and $\bar {c}^{\prime }\cap B^{\prime }=\emptyset $ . Now since , it follows that . But this contradicts the fact that $\psi _{A,B,C}(\bar {x},\bar {y};\bar {z})$ $\dim $ -forks.

Now to prove the claim, let $ \bar {b}^{\prime }_1\subseteq \bar {b}'$ and $ \bar {d}\subseteq N\setminus \operatorname {\mathrm {cl}} (\bar {a}^{\prime })\bar {b}'$ . Notice that implies $\dim (\bar {b}^{\prime }_1/\langle \operatorname {\mathrm {cl}}(\bar {a}^{\prime })\rangle )=\dim (\bar {b}^{\prime }_{1}/\bar {a}^{\prime })$ . Moreover as $\varphi $ is strong, by Definition 2.3 we have $\dim (\bar {b}^{\prime }_{1}/\bar {a}^{\prime })\geq \dim (\bar {b}_{1}/\bar {a})$ . Therefore

$$ \begin{align*} \delta(\bar{b}^{\prime}_1\bar{d}/\langle \operatorname{\mathrm{cl}}(\bar{a}^{\prime})\rangle)&= \dim(\bar{b}^{\prime}_1\bar{d}/\langle \operatorname{\mathrm{cl}}(\bar{a}^{\prime})\rangle)-\alpha |p(\bar{b}^{\prime}_1\bar{d})|\\ &=\dim(\bar{b}^{\prime}_1/\langle \operatorname{\mathrm{cl}}(\bar{a}^{\prime})\rangle)-\alpha |p(\bar{b}^{\prime}_1)|\\ &=\dim(\bar{b}^{\prime}_{1}/\bar{a}^{\prime})-\alpha |p(\bar{b}^{\prime}_1)|\\ &\geq \dim(\bar{b}_{1}/\bar{a})-\alpha |p(\bar{b}_1)|\geq 0.\\[-33pt] \end{align*} $$

Proof Let $M\leqslant N$ be two finitely generated structures in $\mathcal {K}_{\alpha }^{+}$ and $f:M\to \mathbb {M}$ be a strong $\mathcal {L}_{p}$ -embedding. We claim that there is a strong $\mathcal {L}_{p}$ -embedding $g:N\to \mathbb {M}$ extending f.

Since M and N are finitely generated, there are $A\subseteq _{\text {fin}}M$ and $B\subseteq _{\text {fin}}N$ such that $M=\langle A\rangle $ , $N=\langle B\rangle $ and $A\leqslant B\leqslant N$ . By Remark 2.17 we have two specific cases to consider.

Case 1. B is algebraic over A.

Since $A\leqslant N\in \mathcal {K}_{\alpha }^{+}$ , any $x\in B\setminus A$ is non-colored. So by Remark 2.17 it is clear that there is a strong $\mathcal {L}_{p}$ -embedding $g:B\to \mathbb {M}$ extending f.

Case 2. B is transcendental over A. By axiomatization and $\omega $ -saturation there is an embedding $f:B\to \mathbb {M}$ fixing A pointwise and there is no embedding of C over $f(B)$ in $\mathbb {M}$ , for each $C\in \mathcal {C}_{A,B}$ . We claim that $f(B)\leqslant \mathbb {M}$ . Otherwise there should be a tuple $\bar {d}\in \mathbb {M}$ disjoint from $f(B)$ with $f(B)\leqslant _{i}f(B)\bar {d}$ therefore $\delta (\bar {d}/f(B))<0$ . But since all structures in $\mathcal {C}_{A,B}$ are omitted over $f(B)$ , it follows that . So $\delta (\bar {d}/A)=\delta (\bar {d}/f(B))<0$ . But this is a contradiction, since $A\leqslant \mathbb {M}$ .

Proof Clearly any two $\aleph _{0}$ -rich models of $\mathbb {T}_{\alpha }$ are back and forth equivalent. So the above theorem would imply that any $\aleph _{0}$ -saturated models of $\mathbb {T}_{\alpha }$ are back and forth equivalent. Hence $\mathbb {T}_{\alpha }$ is complete.

Fact 4.9. A theory $\Gamma $ is $NTP_{2}$ if and if $ \operatorname {\mathrm {bdn}} (p) < |\Gamma |^{+}$ for every finitary type p if and only if $\operatorname {\mathrm {bdn}}(\text {"}x=x\text {"})<|\Gamma |^{+}$ .

Fact 4.10. Let $\Gamma $ be an arbitrary theory and $\kappa \in $ Card $^{*}$ . Then the following are equivalent.

1. (a) $\operatorname {\mathrm {bdn}}(p)<\kappa $ .

2. (b) If $b \models p(\bar {x})$ and the rows of the array $(\bar {a}_{ij})_{i\in \kappa , j\in \omega }$ are almost mutually indiscernible over A then there are some $i\in \kappa $ and indiscernible sequence $\bar {a}^{\prime }:=(\bar {a}^{\prime }_{j})_{j\in \omega }$ such that $\bar {a}^{\prime }$ is indiscernible over $\bar {b}A$ and $(\bar {a}^{\prime }_{j})_{j\in \omega }\equiv _{\bar {a}_{i0}A}(\bar {a}_{ij})_{j\in \omega }$ .

3. (c) For any mutually indiscernible array $(\bar {a}_{ij})_{i\in \kappa ,j\in \omega }$ over A and $\bar {b}\models p$ , there are some $i \in \kappa $ and indiscernible sequence $\bar {a}^{\prime }:=(\bar {a}^{\prime }_{j})_{j\in \omega }$ such that $\bar {a}^{\prime }$ is indiscernible over $\bar {b}A$ and $(\bar {a}^{\prime }_{j})_{j\in \omega }\equiv _{\bar {a}_{i0}A}(\bar {a}_{ij})_{j\in \omega }$ .

Proof Let $\{\bar {c}_i: i<\kappa \}$ be an $\mathcal {L}_p$ -indiscernible sequence over $\operatorname {\mathrm {cl}}(A)$ that realizes the $EM$ -type of I. This sequence has the same type over A as the type of I, and therefore, there is an $\mathcal {L}_p$ -automorphism $\sigma :\mathfrak {C}\to \mathfrak {C}$ fixing A pointwise such that $\sigma (\bar {c}_i)=\bar {a}_i$ , for each $i<\kappa $ . So, $\{\bar {a}_i: i<\kappa \}$ is $\mathcal {L}_p$ -indiscernible over $\sigma (\operatorname {\mathrm {cl}}(A))=\operatorname {\mathrm {cl}}(A)$ .

Proof We extend the array $I:=(\bar {a}_{ij})_{i\in \kappa ,j\in \omega }$ to a mutually indiscernible array $(\bar {a}_{is})_{i\in \kappa ,s\in \mathbb {Q}}$ over A. Put $Z_{1}:=(\bar {a}_{is})_{i\in \kappa ,s<0 }$ .

It is clear that $(\bar {a}_{ij})_{i\in \kappa ,j\in \omega }$ is mutually indiscernible over $Z_{1}A$ .

To prove that $(\bar {a}_{ij})_{i\in \kappa ,j\in \omega }$ is mutually D-Morley over $Z_{1}A$ , we first show that the row $\bar {a}_{i}$ is a D-Morley sequence over $Z_{1}A$ , for each $i\in \kappa $ . So we prove that , for each $j\in \omega $ . Put $\bar {b}:=(\bar {a}_{il})_{ l<j}$ . Hence it is enough to verify that

Take a finite subset $Z_{0}$ of $Z_{1}A$ with . So the properties of D-independence implies that and . By D-transitivity, to show $(\ast )$ , it is enough to prove that . Put $B=Z_{0}\cap \bar {a}_{i}$ and find $\bar {b}^{\prime }=\{\bar {a}_{il_{0}},\ldots ,\bar {a}_{il_{j-1}}\}$ such that $s<l_{0}<\dots <l_{j-1}<0$ for each s with $\bar {a}_{is}\in B$ . As I is a mutually indiscernible array, $\bar {b}\equiv _{Z_{1}A\bar {a}_{ij}}\bar {b}^{\prime }$ . But since $ \bar {b}^{\prime }\subseteq Z_{1}A$ , it follows that . Hence .

By a similar argument we can prove that the sequence of rows of I is also D-Morley over $Z_{1}A$ . Take $Z=\operatorname {\mathrm {cl}}(Z_{1}A)$ . By Lemma 4.11 we have that the sequence I is also indiscernible over Z.

Proof Let $p(\bar {x})$ be an arbitrary finitary type. We prove that $\operatorname {\mathrm {bdn}}(p)<|\mathbb {T}_{\alpha }|^{+}=\aleph _{1}$ . By Fact 4.10 it is enough to show that for any given mutually indiscernible $I=(\bar {a}_{i})_{i\in \aleph _{1}}$ over $\emptyset $ and $\bar {b}\models p$ , there are some $i \in \aleph _{1}$ and sequence $\bar {a}^{\prime }=(\bar {a}^{\prime }_{j})_{j\in \omega }$ such that $\bar {a}^{\prime }$ is indiscernible over $\bar {b}$ and $(\bar {a}^{\prime }_{j})_{j\in \omega }\equiv _{\bar {a}_{i0}}(\bar {a}_{ij})_{j\in \omega }$ .

By Lemma 4.13 we may find a closed set Z for which $(\bar {a}_{i})_{i\in \aleph _{1}}$ is mutually D-Morley over Z. Since each row $\bar {a}_{i} $ is D-independent over Z, it implies that $\operatorname {\mathrm {cl}}(\bar {a}_{i}Z)=\bigcup _{k\in \omega }\operatorname {\mathrm {cl}}(\bar {a}_{ik}Z)$ . Furthermore for any $i_{1},\ldots ,i_{n}\in \aleph _{1}$ and $j_{1},\ldots ,j_{n}\in \omega $ , we have

$$ \begin{align*} \operatorname{\mathrm{cl}}(\bar{a}_{i_{1}},\dots,\bar{a}_{i_{n}}Z)= \operatorname{\mathrm{cl}}(\bar{a}_{i_{1}}Z)\oplus_{Z}\dots\oplus_{Z}\operatorname{\mathrm{cl}}(\bar{a}_{i_{n}}Z),(\ast) \end{align*} $$

$$ \begin{align*} \operatorname{\mathrm{cl}}(\bar{a}_{i_{1}j_{1}},\dots,\bar{a}_{i_{n}j_{n}}Z)= \operatorname{\mathrm{cl}}(\bar{a}_{i_{1}j_{1}}Z)\oplus_{Z}\dots\oplus_{Z}\operatorname{\mathrm{cl}}(\bar{a}_{i_{n}j_{n}}Z).(\ast\ast) \end{align*} $$

Now by an easy application of Ramsey’s theorem we may assume that the array $J=(\operatorname {\mathrm {cl}}(\bar {a}_{ij}Z))_{i\in \aleph _{1},j\in \omega }$ is mutually indiscernible and mutually D-Morley over Z. Let $\bar {b}_{ij}=\operatorname {\mathrm {cl}}(\bar {a}_{ij}Z)$ .

Take a finite tuple $\bar {e}\models p$ . By D-local character (Fact 4.3) there is a finite subset $I_{0}$ of components of I such that . Choose a countable ordinal $\nu $ such that $I_{0}\subseteq \bigcup _{i<\nu }\bar {a}_{i}$ . Put $I^{\prime }:=(\bar {a}_{ij})_{i>\nu ,j \in \omega }$ , $J^{\prime }=(\bar {b}_{ij})_{i>\nu ,j\in \omega }$ and $\bar {e}^{\prime }=\operatorname {\mathrm {cl}}(\bar {e}I_{0})$ . Then $\bar {e}^{\prime }$ is finite and both $I^{\prime }$ and $J^{\prime }$ are mutually indiscernible and mutually D-Morley over $\operatorname {\mathrm {cl}}(I_{0}Z)$ . Furthermore . Hence by D-closure preservation (item 6 of Fact 4.3) and .

So for any $i>\nu $ and $j_{1}<j_{2}<\dots <j_{k}$ we have that

$$ \begin{align*} \operatorname{\mathrm{cl}}(\bar{b}_{ij_{1}},\dots,\bar{b}_{ij_{k}}I_{0})=\big{(}\bar{b}_{ij_{1}}\oplus_{Z}\cdots\oplus_{Z}\bar{b}_{ij_{k}}\big{)}\oplus_{Z} \operatorname{\mathrm{cl}}(I_{0}Z),(\dagger) \end{align*} $$

$$ \begin{align*} \operatorname{\mathrm{cl}}(\bar{b}_{ij_{1}},\dots,\bar{b}_{ij_{k}}I_{0}\bar{e}^{\prime})= \operatorname{\mathrm{cl}}(\bar{b}_{ij_{1}},\dots,\bar{b}_{ij_{k}}I_{0})\oplus_{\operatorname{\mathrm{cl}}(I_{0})} \bar{e}^{\prime}.(\dagger\dagger) \end{align*} $$

Since T is $NTP_{2}$ and $\bar {e}^{\prime }$ is finite, there are a countable ordinal $t>\nu $ and $\bar {b}^{\prime }_{t}=(\bar {b}^{\prime }_{tj})_{j\in \omega }$ such that $\bar {b}_{t}=(\bar {b}_{tj})_{j\in \omega }\equiv _{\bar {b}_{t0}\operatorname {\mathrm {cl}}(I_{0}Z)}(\bar {b}^{\prime }_{tj})_{j\in \omega }$ and $\bar {b}^{\prime }_{t}$ is an $\mathcal {L}$ -indiscernible over $\operatorname {\mathrm {cl}}(I_{0}Z)\bar {e}^{\prime }$ . Subsequently if we (re-)color $\langle \bar {b}^{\prime }_{t} I_{0}\rangle $ the same as $\langle \bar {b}_{t}I_{0}\rangle $ then we have that $\langle \bar {b}^{\prime }_{t}I_{0}\rangle \in \mathcal {K}_{\alpha }^{+}$ and $\bar {b}_{t0}\operatorname {\mathrm {cl}}(I_{0}Z)\leqslant \mathfrak {C}$ . So $\langle \bar {b}^{\prime }_{t}I_{0}\rangle \oplus _{\langle \bar {b}_{t0}\operatorname {\mathrm {cl}}(I_{0}Z) \rangle }\langle \bar {b}_{t0}\bar {e}^{\prime }I_{0}Z \rangle \in \mathcal {K}_{\alpha }^{+}$ . Now by richness of monster model we may find a closed isomorphic copy $\langle \bar {b}^{\prime \prime }_{t}\rangle $ of $\langle \bar {b}^{\prime }_{t}\rangle $ over $\langle \operatorname {\mathrm {cl}}(I_{0}Z)\bar {e}^{\prime }\rangle $ . Note that since $\bar {b}^{\prime \prime }_{t}$ and $\bar {b}_{t}$ have the same $\mathcal {L}$ -types and colors, $\bar {b}^{\prime \prime }_{t}$ is a $\dim $ -Morley and $\mathcal {L}$ -indiscernible sequence over $\operatorname {\mathrm {cl}}(I_{0}Z)\bar {e}^{\prime }$ . Hence $\bar {b}^{\prime \prime }_{tj_{1}}\ldots \bar {b}^{\prime \prime }_{tj_{n}}=\bar {b}^{\prime \prime }_{tj_{1}}\oplus _{Z}\cdots \oplus _{Z}\bar {b}^{\prime \prime }_{tj_{n}}\leqslant \bar {b}_{t}^{\prime \prime }\leqslant \mathfrak {C}$ , for each $j_{1}<\cdots <j_{n}$ . Furthermore , and ( $\bar {b}^{\prime \prime }_{tj_{1}}\oplus _{Z}\cdots \oplus _{Z}\bar {b}^{\prime \prime }_{tj_{n}})\oplus _{Z} \operatorname {\mathrm {cl}}(I_{0}Z) \leq \mathfrak {C}$ , for each $j_{1}<\cdots <j_{n}$ . Therefore $(\dagger )$ and $(\dagger \dagger )$ also hold for $\bar {b}^{\prime \prime }_{t}$ .

Proof of claim

We prove that for any $i_{1}<\dots <i_{k}$ and $j_{1}<\dots <j_{k}$

(4.1) $$ \begin{align} \langle \operatorname{\mathrm{cl}}(\bar{b}^{\prime\prime}_{ti_{1}},\ldots, \bar{b}^{\prime\prime}_{ti_{k}}I_{0}\bar{e}^{\prime})\rangle \cong_{\langle \operatorname{\mathrm{cl}}(I_{0}Z)\bar{e}^{\prime}\rangle}\langle\operatorname{\mathrm{cl}}(\bar{b}^{\prime\prime}_{tj_{1}},\ldots, \bar{b}^{\prime\prime}_{tj_{k}}I_{0}\bar{e}^{\prime})\rangle. \end{align} $$

Now by $(\dagger )$ and $(\dagger \dagger )$ for $\bar {b}^{\prime \prime }_{t}$ and by Corollary 3.13 in order for $(4.1)$ to hold it is sufficient to have $\operatorname {\mathrm {tp}}_{\mathcal {L}}(\bar {b}^{\prime \prime }_{ti_{1}},\ldots , \bar {b}^{\prime \prime }_{ti_{k}}/\operatorname {\mathrm {cl}}(I_{0}Z)\bar {e}^{\prime })=\operatorname {\mathrm {tp}}_{\mathcal {L}}(\bar {b}^{\prime \prime }_{tj_{1}},\ldots , \bar {b}^{\prime \prime }_{tj_{k}}/\operatorname {\mathrm {cl}}(I_{0}Z)\bar {e}^{\prime })$ . This is because by $(\dagger )$ and $(\dagger \dagger )$ any $\mathcal {L}$ -isomorphism between $\operatorname {\mathrm {cl}}(\bar {b}^{\prime \prime }_{ti_{1}},\ldots , \bar {b}^{\prime \prime }_{ti_{k}}I_{0}\bar {e}^{\prime })$ and $\operatorname {\mathrm {cl}}(\bar {b}^{\prime \prime }_{tj_{1}},\ldots , \bar {b}^{\prime \prime }_{tj_{k}}I_{0}\bar {e}^{\prime })$ which maps $\bar {b}^{\prime \prime }_{ti_{1}},\ldots , \bar {b}^{\prime \prime }_{ti_{k}}$ to $\bar {b}^{\prime \prime }_{tj_{1}},\ldots , \bar {b}^{\prime \prime }_{tj_{k}}$ and fixing $\operatorname {\mathrm {cl}}(I_{0}Z)\bar {e}^{\prime }$ pointwise also preserves the coloring. But the latter equality holds, since $\bar {b}^{\prime \prime }_{t}$ is an $\mathcal {L}$ -indiscernible sequence over $\operatorname {\mathrm {cl}}(I_{0}Z)\bar {e}^{\prime }$ .

Proof To show $\operatorname {\mathrm {bdn}}(\mathbb {T}_{\frac {1}{2}})$ is not finite, in the light of item $(b)$ of Fact 4.10, for each natural number k we must construct an almost $\mathcal {L}_{p}$ -mutually indiscernible array $(a_{ij})_{i\in k, j\in \omega }$ over $\emptyset $ and find $b^{\prime }$ such that for each $i\in k$ and for each $\mathcal {L}_{p}$ -indiscernible sequence $\bar {a}^{\prime }:=(a^{\prime }_{j})_{j\in \omega }$ with $\bar {a}^{\prime }$ is indiscernible over $b^{\prime }$ we have $(a^{\prime }_{j})_{j\in \omega }\not \equiv _{a_{i0}}(a_{ij})_{j\in \omega }$ .

Let $b\in \mathfrak {C}$ be non-algebraic over $\emptyset $ . Choose elements $\{b_{1},\dots ,b_{k}\}$ of $\mathfrak {C}$ such that $\{b_{1},\dots ,b_{k}\}$ is independent over b. Now by Lemma 2.10 we may choose $b_{k+1},\dots ,b_{2k+1}\in \operatorname {\mathrm {acl}}(b,b_{1},\dots ,b_{k})$ such that any k-element subset of $\{b_{1},\dots ,b_{2k+1}\}$ is a base for $\operatorname {\mathrm {acl}}(b,b_{1},\dots ,b_{2k+1})=\operatorname {\mathrm {acl}}(b,b_{1},\dots ,b_{k})$ . Let $c_{i}=b_{k+2+i}$ for each $0\leq i\leq k-1$ . Hence $C=\{c_{0},\dots ,c_{k-1}\}$ is independent over $\emptyset $ . Choose an $\mathcal {L}$ -indiscernible sequence $\{c_{0j}:j\in \omega \}$ over $\{c_{1},\dots ,c_{k-1}\}$ with $c_{00}=c_{0}$ . Respectively by induction over $i\in k$ , choose an $\mathcal {L}$ -indiscernible sequence $\{c_{ij}: j\in \omega \}$ over $\bar {c}_{<i}(c_{l0})_{l>i}$ with $c_{i0}=c_{i}$ . Since $\{c_{0},\dots ,c_{k-1}\}$ is independent over $\emptyset $ , by Ramsey theorem and compactness there exists an array $I:=(c_{ij})_{i\in k, j\in \omega }$ such that $c_{ij}$ ’s are distinct. Furthermore by free amalgamation of T and Observation 2.8 we may choose I in such a way that . It is easy to see that I is an almost $\mathcal {L}$ -mutually indiscernible sequence over $\emptyset $ .

Now we color $\langle I b b_{1}\ldots b_{k+1}\rangle $ by letting the elements of $I\cup \{b,b_{1},\ldots ,b_{k+1}\}$ colored while leaving the rest of elements non-colored. By exchange property of T and since I is almost mutually indiscernible, each row $\bar {c}_{i}$ is $\dim $ -independent from $\bar {c}_{\neq i}$ . So for each subset X of $\bar {c}_{i}$ we have that $X\leq I$ . Moreover since , we have $I\leq Ib b_{1}\ldots b_{k+1}$ . Therefore $\langle I b b_{1}\dotsb _{k+1}\rangle \in \mathcal {K}_{\alpha }^{+}$ . Take $f:\langle I b b_{1}\ldots b_{k+1}\rangle \to \mathfrak {C}$ to be a strong $\mathcal {L}_{p}$ -embedding with $b^{\prime }=f(b)$ , $b^{\prime }_{t}=f(b_{t})$ and $a_{ij}=f(c_{ij})$ for each $t\in k+2$ , $i\in k$ and $j\in \omega $ . Then it is clear that $(a_{ij})_{i\in k,j\in \omega }$ is an almost $\mathcal {L}_p$ -mutually indiscernible array over $\emptyset $ . Further we have that,

$$ \begin{align*} \delta(a_{00},\ldots ,a_{(k-1) 0},b^{\prime}_{1},\ldots ,b^{\prime}_{k+1}/b^{\prime})=k-\frac{1}{2}(2k+1)=-\frac{1}{2}. \end{align*} $$

So $(b^{\prime }, \{b^{\prime }, a_{00},\ldots ,a_{(k-1) 0},b^{\prime }_{1},\ldots ,b^{\prime }_{k+1}\})$ is a minimal pair and hence by Fact 2.15 we have that $a_{i0}\in \operatorname {\mathrm {cl}}(b^{\prime })$ for each $i\in k$ . We claim that for each $i\in k$ and $\mathcal {L}_{p}$ -indiscernible sequence $(a_{j}^{\prime })_{j\in \omega }$ over $b^{\prime }$ we have that $(a^{\prime }_{j})_{j\in \omega }\not \equiv _{a_{i0}}(a_{ij})_{j\in \omega }$ .

Assume not. Then for some $i\in k$ we have $(a^{\prime }_{j})_{j\in \omega }\equiv _{a_{i0}}(a_{ij})_{j\in \omega }$ . Note that since $a^{\prime }_{0}=a_{i0}\in \operatorname {\mathrm {cl}}(b^{\prime })$ and $(a^{\prime }_{j})_{j\in \omega }$ is an $\mathcal {L}_{p}$ -indiscernible sequence over $b^{\prime }$ , we have that $a^{\prime }_{j}\in \operatorname {\mathrm {cl}}(b^{\prime })$ for each $j\in \omega $ .

But this gives infinitely many elements in $\operatorname {\mathrm {cl}}(b^{\prime })$ which contradicts the fact that $\operatorname {\mathrm {cl}}(b^{\prime })$ is finite.

Fact 4.19. Assume there is an $Aut(\mathfrak {M})$ -invariant ternary relation on small subsets of the monster $\mathfrak {M} \models \Gamma $ which satisfies the following properties. For an arbitrary $M \models \Gamma $ and arbitrary tuples and subsets from $\mathfrak {M}$ ,

1. 1. Strong finite character: If , then there is a formula $ \varphi (\bar {x},\bar {b},\bar {m})\in tp(\bar {a}/BM)$ such that for any $\bar {a}^{\prime }\models \varphi (\bar {x},\bar {b},\bar {m})$ , .

2. 2. Existence over models: $M \models \Gamma $ implies for any $\bar {a}\in \mathfrak {M}$ .

3. 3. Monotonicity: If then .

4. 4. Symmetry: if and only if .

5. 5. The independence theorem: , , and $\bar {a} \equiv _{M} \bar {a}^{\prime }$ implies there is $\bar {a}^{\prime \prime }$ with $\bar {a}^{\prime \prime } \equiv _{MB} \bar {a}$ , $\bar {a}^{\prime \prime } \equiv _{MC} \bar {a}^{\prime }$ and .

Then $\Gamma $ is $NSOP_{1}$ .

Proof It is enough to prove that satisfies the conditions of Fact 4.19. It is easily seen that is an automorphism invariant and satisfies the properties existence over models, monotonicity, and symmetry by the definition of .

So we only prove the other properties. Notice that without loss of generality $B,B^{\prime }$ and $ C$ can be considered to be closed in $\mathfrak {C}$ .

• • Strong finite character: Assume that . We prove that there is a formula $ \varphi (\bar {x},\bar {b},\bar {m})\in \operatorname {\mathrm {tp}}(\bar {a}/BM)$ such that for any $\bar {a}^{\prime }\models \varphi (\bar {x},\bar {b},\bar {m})$ , .

  implies either or . Then we have two cases to consider:

  Case 1. but $\operatorname {\mathrm {cl}}(\bar {a}M)\cap \operatorname {\mathrm {cl}}(BM)=M$ .

  So $D(\bar {a}/M)>D(\bar {a}/MB)$ . Therefore there exist a finite set $M_{0}\leqslant M$ and a finite tuple $\bar {b}$ of B such that

  $$ \begin{align*} D(\bar{a}/M)=D(\bar{a}/M_{0})>D(\bar{a}/MB)=D(\bar{a}/M_{0}\bar{b}). \end{align*} $$
  By D-symmetry this is equivalent to
  $$ \begin{align*} D(\bar{b}/M_{0})>D(\bar{b}/M_{0}\bar{a}).(\ast) \end{align*} $$
  Let $\gamma =D(\bar {b}/M_{0})$ . Now in the light of Lemma 4.5 there exists a partial type $\Sigma _{\gamma }(\bar {x},\bar {y})$ over M expressing that $D(\bar {y}/M\bar {x})\geq \gamma $ . Hence by $(\ast )$ one can find a formula $\theta (\bar {x},\bar {m}_{0},\bar {b})$ which is satisfied by $\bar {a}$ and
  $$ \begin{align*} D(\bar{b}/M)=D(\bar{b}/M_{0})>D(\bar{b}/M\bar{a}^{\prime}), \end{align*} $$
  for every $\bar {a}^{\prime }$ with $\theta (\bar {a}^{\prime },\bar {m}_{0},\bar {b})$ . Hence by D-symmetry
  $$ \begin{align*} D(\bar{a}^{\prime}/M)>D(\bar{a}^{\prime}/M\bar{b}). \end{align*} $$
  Thus .

  Case 2. $\operatorname {\mathrm {cl}}(\bar {a}M)\cap \operatorname {\mathrm {cl}}(BM)\neq M$ or .

  By Remark 4.20 this case is equivalent to .

  Let . Since T is $NSOP_{1}$ , there are $\bar {a}^{\prime }\subseteq \operatorname {\mathrm {cl}}(\bar {a}M)$ , disjoint from $\bar {a}$ , and an $\mathcal {L}$ -formula $\theta (\bar {x},\bar {x}^{\prime },\bar {b},\bar {m})\in \operatorname {\mathrm {tp}}_{\mathcal {L}}(\bar {a}\bar {a}^{\prime }/\operatorname {\mathrm {cl}}(BM))$ witnessing the strong finite character property with respect to T. Let $\psi (\bar {x},\bar {x}^{\prime },\bar {m})$ be an $\mathcal {L}_{p}$ -intrinsic formula satisfied by $\bar {a}\bar {a}^{\prime }$ . Set $\varphi (\bar {x},\bar {b},\bar {m}):= \exists \bar {x}^{\prime } ( \theta (\bar {x},\bar {x}^{\prime },\bar {b},\bar {m} )\wedge \psi (\bar {x},\bar {x}^{\prime },\bar {m}))$ . We claim that $\varphi (\bar {x},\bar {b},\bar {m})$ is the desired formula. Let $\bar {a}_{1}$ be a solution of $\varphi (\bar {x},\bar {b},\bar {m})$ . Then there exits $\bar {a}_{2}\subseteq \operatorname {\mathrm {cl}}(\bar {a}_{1}M)$ with $\theta (\bar {a}_{1},\bar {a}_{2},\bar {b},\bar {m})$ . Hence . Therefore .

• • The independence theorem: Assume that , , and $\bar {a} \equiv _{M} \bar {a}^{\prime }$ . Further, suppose that $B=\langle B\rangle ,C=\langle C \rangle $ , both include M and they are closed in $\mathfrak {C}$ . We prove that there is $\bar {a}^{\prime \prime }$ with $\bar {a}^{\prime \prime } \equiv _{B} \bar {a}$ , $\bar {a}^{\prime \prime } \equiv _{C} \bar {a}^{\prime }$ and . By definition of we have , , and $\bar {a}\equiv _{M} \bar {a}^{\prime }$ (in the sense of $\mathcal {L}_{p}$ ). Since $\bar {a}\equiv _{M} \bar {a}^{\prime }$ , by Corollary 3.13 it follows that $\langle \operatorname {\mathrm {cl}}(\bar {a}M)\rangle \equiv _{M}\langle \operatorname {\mathrm {cl}}(\bar {a}^{\prime }M)\rangle $ .

  So by independence theorem for T there exists an $\mathcal {L}$ -structure E such that $\operatorname {\mathrm {tp}}_{\mathcal {L}}(E/B)=\operatorname {\mathrm {tp}}_{\mathcal {L}}(\langle \operatorname {\mathrm {cl}}(\bar {a}M)\rangle /B)$ , $\operatorname {\mathrm {tp}}_{\mathcal {L}}(E/C)=\operatorname {\mathrm {tp}}_{\mathcal {L}}(\langle \operatorname {\mathrm {cl}}(\bar {a}^{\prime }M)\rangle /C)$ and . By Remark 4.20 we have . Existence of an $\mathcal {L}$ -isomorphism $f:\langle \operatorname {\mathrm {cl}}(\bar {a}M) \rangle \to E$ allows us to color E the same as $\langle \operatorname {\mathrm {cl}}(\bar {a}M)\rangle $ . So $E\in \mathcal {K}_{\alpha }^{+}$ .

  Let $D=\langle B C\rangle $ and $F= E\oplus _{M} D$ be an $\mathcal {L}$ -structure given according to Convention 2.7. We subsequently turn F to an $\mathcal {L}_{p}$ -structure by taking $p(F)=p(E)\cup p(D)$ . Then by Lemma 3.3 we have that $D\leqslant F\in \mathcal {K}_{\alpha }^{+}$ . Therefore there is a strong embedding $g:F\to \mathfrak {C}$ fixing D pointwise. Put $g(f(\bar {a}))=\bar {a}^{\prime \prime }$ . It can be easily seen that $\bar {a}^{\prime \prime } \equiv _{B} \bar {a}$ , $\bar {a}^{\prime \prime } \equiv _{C} \bar {a}^{\prime }$ (in the sense of $\mathcal {L}_{p}$ )and . Thus . Furthermore $\operatorname {\mathrm {cl}}(\bar {a}^{\prime \prime }BC)=\operatorname {\mathrm {cl}}(\bar {a}^{\prime \prime }M)\oplus _{M}BC$ . Therefore in the light of Lemma 4.4 we have and consequently .

Fact 4.24. Let $\Gamma $ be an arbitrary theory, and be an $Aut(\mathfrak {M})$ -invariant ternary relation on small subsets of the monster model $\mathfrak {M} \models \Gamma $ . Suppose has the following properties.

1. 1. Local character: For any $\bar {a}$ and $ B$ there is $A\subseteq B$ such that the cardinality of A is at most the cardinality of $\Gamma $ and .

2. 2. Finite character: if and only if for every finite tuple $\bar {b}$ from B, .

3. 3. Extension: For any $\bar {a},A$ and $B \supseteq A$ there is $\bar {a}^{\prime }$ such that $\operatorname {\mathrm {tp}}(\bar {a}^{\prime }/A) = \operatorname {\mathrm {tp}}(\bar {a}/A)$ and .

4. 4. Symmetry: If then .

5. 5. Transitivity: Suppose $A \subseteq B \subseteq C$ . Then and if and only if .

6. 6. The independence theorem: Suppose M is a model of $\Gamma $ , , , and $\bar {a} \equiv _{M} \bar {a}^{\prime }$ . Then there is $\bar {a}^{\prime \prime }$ with $\bar {a}^{\prime \prime } \equiv _{MB} \bar {a}$ , $\bar {a}^{\prime \prime } \equiv _{MC} \bar {a}^{\prime }$ , and .

Then is exactly the forking independence and $\Gamma $ is simple.

Proof As T is simple there is a notion of independence which satisfies the properties of Fact 4.24. Furthermore since is the T-forking independence, we may assume that and, furthermore implies .

Let be the notion of independence introduced in Definition 4.21. To simplify our discussion sometimes is used as an abbreviation of . We prove that satisfies the conditions 1–6 of Fact 4.24.

One can easily prove invariance, monotonicity, symmetry, and finite character. Further the proof of independence theorem is the same as of Theorem 4.22. So we prove the other properties.

• • Local character. Let $\bar {a}$ and a closed subset B be given. Take a finite closed subset $B_{0}$ of $ B$ such that . So by D-closure preservation of Fact 4.3, we have . Now by local character property of there exists a countable closed subset $B_{1}$ of $ B$ including $B_{0}$ such that and $\operatorname {\mathrm {cl}}(\bar {a} B_{1})\cap B=B_{1}$ . Therefore . By iterating this process a sequence $B_{0}\leqslant B_{1}\leqslant B_{2}\leqslant \dots $ is found such that and for each $i\in \mathbb {N}$ . Now put $A=\bigcup _{i\in \mathbb {N}}B_{i}$ . Hence is obtained.

• • Extension. Let $\bar {a}, A$ and $B \supseteq A$ be given. Without loss of generality we may assume that $A\leqslant B\leqslant \mathfrak {C}$ . By simplicity of T and the extension property of there are $B_{0}$ and a partial $\mathcal {L}$ -isomorphism $f: \operatorname {\mathrm {cl}}(\bar {a}A)\to B_{0}$ fixing A such that . Now by taking $\bar {b}=f(\bar {a})$ (and coloring $B_{0}$ using f and leaving $B_{0}-Im(f)$ non-colored) the function f can be considered as a partial $\mathcal {L}_{p}$ -isomorphism between $\operatorname {\mathrm {cl}}(\bar {a}A)$ and $\operatorname {\mathrm {cl}}(\bar {b}A)$ .

  Since implies we have . Let $C= B\oplus _{A} B_{0}$ . Then $B_{0}\leqslant C$ . Take $g:C\to \mathfrak {C}$ to be a strong embedding and $\bar {a}^{\prime }=g(\bar {b})$ . Then we have .

• • Transitivity. Suppose $A \subseteq B \subseteq C$ . We show that and if and only if .Fix $A \leqslant B \leqslant C\leqslant \mathfrak {C}$ . Let and . So by the definition of we have and . We prove that .

  Since and , Lemma 4.4 implies $\operatorname {\mathrm {cl}}(\bar {a}C)=\operatorname {\mathrm {cl}}(\bar {a}B)\oplus _{B} C$ and $\operatorname {\mathrm {cl}}(\bar {a}B)=\operatorname {\mathrm {cl}}(\bar {a}A)\oplus _{A} B$ . Hence $\operatorname {\mathrm {cl}}(\bar {a}C)=\operatorname {\mathrm {cl}}(\bar {a}A)\oplus _{A} C$ and .

  Then $\operatorname {\mathrm {cl}}(\bar {a}A)\subseteq \operatorname {\mathrm {cl}}(\bar {a}B)$ yields . Now as we have , by transitivity of , is obtained

  Conversely assume that . Then . We prove that and .

  By transitivity of and it is clear that and . On the other hand since , by Lemma 4.4 we have $\operatorname {\mathrm {cl}}(\bar {a}B)=\operatorname {\mathrm {cl}}(\bar {a}A)\oplus _{A} B$ . Hence is equivalent to . But by symmetry, transitivity of and the latter holds.

Proof Since T is supersimple, by Proposition 5.5 of [Reference Ben-Yacov and Chernikov8], it is strong. Hence by Theorem 4.16 the theory $\mathbb {T}_{\frac {1}{2}}$ is a strong simple theory of unbounded weight.