Recently NSOP
${}_1$
theories have been significantly studied mainly due to that in NSOP
${}_1$
theories, nice properties, such as symmetry, transitivity, and the Independence Theorem, hold over models with Kim-independence, as proved in [Reference Kaplan and Ramsey4, Reference Kaplan and Ramsey5]. Later in [Reference Chernikov, Kim and Ramsey1, Reference Dobrowolski, Kim and Ramsey3], the results are extended ‘over arbitrary sets’ in any NSOP
${}_1$
theory with existence. The class of NSOP
${}_1$
theories is considered as the proper superclass of that of simple theories, where parallel theoretic development to that on simple theories is expected to be made. In particular all simple theories have existence. Therefore it is a burning question to know whether every NSOP
${}_1$
theory has existence.Footnote
1
We investigate this problem in this article and achieve a partial success. Namely we show that Kim-forking satisfies existence (over all sets) in all NSOP
${}_1$
theories. More precisely, the failure of existence cannot be witnessed by Morley sequences, in all NSOP
${}_1$
theories. This implies that any complete type in NSOP
${}_1 T$
has 
-Morley sequences in it. Hence in such a theory, it makes sense to define a new kind of ‘dividing’ of a formula witnessed by such a sequence (which is known to be equivalent to the usual ‘Kim-dividing’ when the theory has existence), although we do not at all intend to push forward its study in this note.
In Section 1, we collect basic definitions and facts necessary for the rest of the article. In Section 2, preliminary lemmas playing key roles in proving the main theorem are presented. In Section 3, we state and prove our main theorem that Kim-forking satisfies existence in all NSOP
${}_1 T$
. We will discuss further related issues in Section 4.
1. Introduction
In this section we summarize necessary definitions and facts (from [Reference Chernikov and Ramsey2–Reference Kim6], etc.) which will be used freely throughout the article. We work with a complete theory T in a language
${\mathcal L}$
with its large model
${\mathcal M}$
. As usual, (small) sets
$A, B, C, \dots $
and (possibly infinite) tuples
$a_i, b_i,\dots $
are from
${\mathcal M}$
and models
$M, N,\dots $
are small elementary submodels of
${\mathcal M}$
. Given a sequence
$(A_i\mid i<\kappa )$
and
$j'<j<\kappa $
, in addition to
$A_{<j}$
,
$A_{\geq j}$
, and so on, we use
$A_{j'<<j}$
,
$A_{\ne k}$
to denote
$(A_k\mid j'<k<j)$
,
$(A_i\mid i(\ne k)\in \kappa )$
, respectively.
We begin to recall basic definitions related to dividing and forking. We also recall the notion of a Morley sequence over
$(A,B)$
originated from [Reference Shelah10], which is useful in developing our arguments.
Definition 1.1.
-
(1) We say a formula
$\varphi (x,a_0)$
divides over A if there is an A-indiscernible sequence
$\langle a_i\mid i<\omega \rangle $
such that
$\{\varphi (x,a_i)\mid i<\omega \}$
is inconsistent. A formula forks over A if the formula implies some finite disjunction of formulas, each of which divides over A. Then we say a type divides/forks over A if the type implies a formula which divides/forks over A. -
(2) We write
if
$\operatorname {tp}(A/BC)$
does not fork over B. A sequence
$I=\langle a_i\mid i<\omega \rangle $
is said to be -independent (or just independent) over A (or A-independent) if for each
$i<\omega $
; if additionally
$a_i\equiv _A a_0$
then we say it is A-independent in
$\operatorname {tp}(a_0/A)$
. We say I is independent over
$(A,B)$
if for all
$i\in \omega $
. -
(3) An A-independent sequence
$I=\langle a_i\mid i<\omega \rangle $
is said to be Morley over A (or A-Morley) if it is A-indiscernible. When I is Morley over A, we also call it a Morley sequence in
$\operatorname {tp}(a_0/A)$
. I is said to be Morley over
$(A,B)$
if it is independent over
$(A,B)$
and I is
$AB$
-indiscernible.
if 
-independent (or just independent) over A (or A-independent) if 
for each 
for all 
We now point out the following handy fact which we will use frequently. A proof can be found, say in [Reference Kim6].
Fact 1.2. Given
$\kappa $
, there is
$\lambda = \lambda (\kappa )$
such that for any sequence
$J=\langle a_i: i<\lambda \rangle $
of
$\leq \kappa $
-sized tuples and any set A of size
$\leq \kappa $
, there is an A-indiscernible sequence
$\langle b_i: i<\omega \rangle $
which is finitely based on J over A, i.e., for any
$b_{\leq n}$
there are
$a_{i_{\leq n}}$
with
$i_0<\dots <i_n\in \lambda $
such that
$ b_{\leq n}\equiv _A a_{i_{\leq n}}$
.
Straightforward consequences from the definitions of dividing and forking are summarized as follows.
Fact 1.3.
-
(1) (Finite character)
iff for any finite
$a\in A$
,
$c\in C$
we have . -
(2) (Base monotonicity) If
holds then follows. Hence if I is Morley over
$(A,B)$
then it is Morley over
$AB$
as well. -
(3) (Left transitivity) If
and then we have . -
(4) (Extension) For
$A\subseteq B\subseteq C$
, if then there is
$D'\equiv _{B}D$
such that . -
(5) Using extension and Fact 1.2, the equivalence of the following is obtained.
-
(a) Any type over a set does not fork over the set.
-
(b) For any
$p\in S(A)$
there is a Morley sequence in p. -
(c) Any consistent formula
$\varphi (x,a)$
does not fork over its finite parameter a.
-
-
(6) (Existence) We say T has existence if one of the equivalent conditions in (5) holds. Notice that
holds for every model
$M.$
Hence given any complete type over a model, there is a Morley sequence in it.
iff for any finite 
.
holds then 
follows. Hence if I is Morley over 
and 
then we have 
.
then there is 
.
holds for every model 
As is well-known a typical example where existence (over
$\emptyset $
) fails to hold is a circle with a ternary relation R on the circle such that
$R(a,b, c)$
holds iff
$a,c$
are not antipodal and b is in the smaller arc connecting a and c. We will discuss this example again in Section 4.
Definition 1.4.
-
(1) A formula
$\varphi (x,a_0)$
Kim-divides over A if there is an A-Morley sequence
$\langle a_i\mid i<\omega \rangle $
such that
$\{\varphi (x,a_i)\mid i<\omega \}$
is inconsistent. A formula Kim-forks over A if the formula implies some finite disjunction of formulas, each of which Kim-divides over A. A type Kim-forks over A if the type implies a formula which Kim-forks over A. -
(2) We write
if
$\operatorname {tp}(A/BC)$
does not Kim-fork over B. Obviously implies .A sequence
$I=\langle a_i\mid i<\omega \rangle $
is said to be -independent over A if for each
$i<\omega $
; if additionally
$a_i\equiv _A a_0$
then we say it is -independent (over A) in
$\operatorname {tp}(a_0/A)$
. -
(3) Given an A-indiscernible sequence
$\langle a_i\mid i<\omega \rangle $
we say it is
-Morley over A (or A-
-Morley) if it is -independent over A; and we say it is a total
-Morley sequence over A if holds for each
$i<\omega $
. Due to base monotonicity and left transitivity, a Morley sequence over A is a total Morley sequence over A, so a total -Morley sequence over A as well.
if 
implies 
.
-independent over A if 
for each 
-independent (over A) in 
-Morley over A (or A-
-Morley) if it is 
-independent over A; and we say it is a total 
-Morley sequence over A if 
holds for each 
-Morley sequence over A as well. In the definitions above if
$A=\emptyset $
then we omit to state it. For example ‘
$\varphi (x,a)$
(Kim-)divides/forks’ means it does over
$\emptyset $
.
Remark 1.5.
-
(1) Since we do not assume existence in Definition 1.4, given formulas
$\varphi (x,a)\models \psi (x,b)$
, it is unclear whether the following property holds: if
$\psi (x,b)$
Kim-divides over A then so does
$\varphi (x,a)$
over A. However this property holds with respect to Kim-forking (note that if
$\varphi (x,a)\models \psi (x,b)$
and
$\psi $
implies a finite disjunction of Kim-dividing formulas, then
$\varphi $
also implies it). In particular, if
$p(x)\models q(x)$
and
$q(x)$
Kim-forks over A then so does
$p(x)$
over A; Kim-forking of a type over A only depends on its solution set; and a type
$p(x)$
Kim-forks over A iff
$p(x)$
implies some finite disjunction of formulas, each of which Kim-divides over A. -
(2) (Finite character for
) iff for any finite
$a\in A$
,
$c\in C$
we have . -
(3) (Extension for
) By the same compactness argument we have the following: If a partial type p over C does not Kim-fork over a set then there is a completion of p over C which does not Kim-fork over the set. In particular, for
$A\subseteq B\subseteq C$
, if then there is
$a'\equiv _{B}a$
such that . -
(4) By the same techniques (using (3) and Fact 1.2) the equivalence of the following can be obtained.
-
(a) Any type over a set does not Kim-fork over the set.
-
(b) For any
$p\in S(A)$
there is an -Morley sequence in p.
We say Kim-forking has existence if one of the above equivalent conditions holds.
-
) 
iff for any finite 
.
) By the same compactness argument we have the following: If a partial type p over C does not Kim-fork over a set then there is a completion of p over C which does not Kim-fork over the set. In particular, for 
then there is 
.
-Morley sequence in p. Now we recall a theory being (N)SOP
${}_1$
, or simple.
Definition 1.6. We say a formula
$\varphi (x,y)\in {\mathcal L}$
has SOP
$_{1}$
if there is a tree of tuples
$\{a_{\eta }\mid \eta \in 2^{<\omega }\}$
such that:
-
(1) for all
$\xi \in 2^{\omega }$
,
$\{\varphi (x,a_{\xi \lceil \alpha }) \mid \alpha < \omega \}$
is consistent; -
(2) for all
$\eta \in 2^{<\omega }$
, if
$\nu \unrhd \eta ^\smallfrown \langle 0 \rangle $
, then
$\{\varphi (x;a_{\nu }), \varphi (x;a_{\eta ^\smallfrown \langle 1\rangle })\}$
is inconsistent,
where
$\unlhd $
denotes the tree partial order on
$2^{<\omega }$
. We say T has SOP
$_{1}$
if some formula has SOP
$_{1}$
in T, and say T is NSOP
$_{1}$
if T does not have SOP
${}_1$
.
Fact 1.7. [Reference Chernikov and Ramsey2, Reference Kaplan and Ramsey4] T has SOP
${}_1$
iff there is a formula
$\varphi (x,y)$
and an indiscernible sequence
$(b_ic_i\mid i<\omega )$
such that for each
$i<\omega $
,
$b_i\equiv _{(bc)_{<i} }c_i$
(*); and
${\{\varphi (x, b_i)\mid i\in \omega \}}$
is consistent; while
$\{\varphi (x, c_i)\mid i\in \omega \}$
is inconsistent. (Notice that by compactness the same holds even if we replace (*) by
$b_i\equiv _{(bc)_{>i} }c_i$
.)
We say T is simple if 
satisfies local character: For any set A and finite a, there is
$A_0\subseteq A$
with
$|A_0|\leq |T|$
such that 
. As is well known, in simple T, existence holds; and dividing, forking, and Kim-forking are all equal over any set. Any simple T is NSOP
${}_1$
. It is not known yet whether existence holds in all NSOP
${}_1$
theories. In this note, however, we show that Kim-forking satisfies existence in every NSOP
${}_1$
theory. We will only use the following two facts holding in any NSOP
${}_1$
theory.
Fact 1.8. [Reference Kaplan and Ramsey4] Assume T is NSOP
${}_1$
.
-
(1) A formula Kim-divides over a model M iff the formula Kim-forks over M. Moreover,
for any model M (since ). -
(2) (Kim’s lemma for Kim-dividing over a model) A formula
$\varphi (x,a_0)$
Kim-divides over a model M iff for any Morley sequence
$\langle a_i\mid i<\omega \rangle $
over M,
$\{\varphi (x,a_i)\mid i<\omega \}$
is inconsistent.
for any model M (since 
).
We state the following folklore which comes mainly from extension.
Fact 1.9. Given a cardinal
$\kappa $
, let
$I=\langle a_i\mid i<\kappa \rangle $
be 
-independent over
$(A,B)$
.
-
(1) For any C there is
$C'\equiv _{AB} C$
such that I is independent over
$(A,BC')$
. It follows that and I is
$ABC'$
-independent. -
(2) Suppose additionally
$\kappa $
is infinite and I is Morley over
$(A,B)$
. Then again for any C, there is
$C'\equiv _{AB}C$
such that I is Morley over
$(A,BC')$
.
and I is 
Proof. (1) It suffices to verify the first statement, since it implies the second one due to base monotonicity and left transitivity. Given
$i<\kappa $
, inductively assume that we have found
$(b_j\mid j<i)$
such that
$b_{<i}\equiv _{AB}a_{<i}$
and
$(b_j\mid j<i)$
is independent over
$(A, BC)$
(*). It is enough (due to an
$AB$
-automorphism) to find
$b_i$
such that
$b_{\leq i}\equiv _{AB}a_{\leq i}$
and
$(b_j\mid j\leq i)$
is independent over
$(A, BC)$
. Now by (*), there clearly is
$b^{\prime }_i$
such that
$b_{<i}b^{\prime }_i\equiv _{AB} a_{<i} a_i $
, so that 
. Then by extension there is
$b_i\equiv _{Ab_{<i}B} b^{\prime }_i$
such that 
. As wanted,
$b_{\leq i}\equiv _{AB}b_{<i}b^{\prime }_i\equiv _{AB}a_{\leq i}$
and
$b_{\leq i}$
is independent over
$(A,BC)$
.
(2) Due to compactness, we can stretch out
$I_\omega :=\langle a_i\mid i<\omega \rangle (\subseteq I)$
to
$I_\lambda =\langle a_i\mid i<\lambda \rangle $
for some sufficiently large
$\lambda $
(to apply Fact 1.2), which is still Morley over
$(A, B)$
. Then by (1), there at least is
$C_0\equiv _{AB}C$
such that
$I_\lambda $
is independent over
$(A, BC_0)$
. Now we apply Fact 1.2 to obtain
$ABC_0$
-indiscernible
$J=(b_i\mid i<\omega )$
finitely based on
$I_\lambda $
over
$ABC_0$
, so that clearly J is Morley over
$(A, BC_0)$
. Notice that
$J\equiv _{AB} I_{\omega }$
, since
$I_\omega $
is
$AB$
-indiscernible. Hence there is
$C_1$
such that
$C_1\equiv _{AB}C_0$
and
$I_\omega $
is Morley over
$(A, BC_1)$
. Again by compactness we can extend
$I_\omega $
to
$I'$
of length
$\kappa $
such that
$I'$
is Morley over
$(A, BC_1)$
. It also follows
$I\equiv _{AB}I'$
and by an
$AB$
-automorphism we find a desired
$C'\equiv _{AB}C_1$
such that the original I is Morley over
$(A, BC')$
.
Throughout this note we give the lexicographic order to
$\mu \times \lambda $
so that for an array
$(c_{ij}\mid (i,j)\in \mu \times \lambda )$
,
$c_{<ij}$
means
$\{c_{i'j'}\mid i'<i \text {, or } (i'=i \text { and } j'<j ) \}$
, and so on.
Definition 1.10. Assume an array
$I=(c_{ij}\mid i,j\in \omega )$
is given with
$I_i:=(c_{ij}\mid j\in \omega )$
.
-
(1) Recall that I is said to be strongly indiscernible over A if
$(I_i\mid i<\omega )$
is A-indiscernible and each
$I_i$
is
$AI_{\ne i}$
-indiscernible. -
(2) I is said to be a Morley array over A if it is strongly indiscernible over A, and
holds for each
$(i,j)\in \omega \times \omega $
. -
(3) I is a weak* Morley array over A if
holds for each
$(i,j)\in \omega \times \omega $
, each
$I_i$
is A-indiscernible; and if additionally
$(I_i\mid i\in \omega )$
is an A-Morley sequence then we call I a weak Morley array over A.
holds for each 
holds for each 
If I is a Morley array over A, then mainly by left transitivity and base monotonicity, I is a weak Morley array.
2. Preliminaries
In this section we will present a couple of key lemmas critical in showing the main result Theorem 3.2. Sophisticated applications of array and tree indiscernibilities keeping 
-independence enable us to prove those lemmas.
Lemma 2.1. Assume that
$I=(c_{ij}\mid i,j\in \omega )$
is an A-Morley array. Then for any
$(A\subseteq )B$
, there is
$I'\equiv _AI$
such that
$I'$
is a weak B-Morley array.
Proof. We choose
$\mu ,\kappa $
to be the cardinals
$\lambda (|B|+|I|)$
and
$\lambda (\mu )$
, respectively, as described in Fact 1.2. By compactness we can stretch out I to
$(c_{ij}\mid i\in \mu , \ j\in \kappa )$
while keeping strong indiscernibility over A. Then it must be A-Morley as well (*). By induction on
$\mu $
we will construct the following sequences.
Claim 2.1.1. For each
$i<\mu $
there is
$J_i=(c^{\prime }_{mj}\mid m\leq i;\ j\in \omega )$
such that:
-
(i)
$J_i\equiv _A(c_{mj}\mid m\leq i;\ j\in \omega )$
and for
$i'<i$
,
$J_{i'}\subseteq J_i$
; -
(ii)
$J_i$
is a weak* B-Morley array.
Proof of Claim 2.1.1.
Given
$i<\mu $
, inductively assume that for all
$i'<i$
, we have found
$J_{i'}$
satisfying (i) and (ii). Now put
$J_{<i}:=\bigcup _{i'<i} J_{i'}=(c^{\prime }_{mj}\mid m<i,\ j<\omega )$
(so if i is a successor then
$J_{<i}=J_{i-1}$
). Then due to (i), there is
$J^{\prime }_i=(d^{\prime }_{ij}\mid j<\kappa )$
such that
$J_{<i}J^{\prime }_i\equiv _A(c_{mj}\mid m< i,\ j\in \omega )(c_{ij}\mid j\in \kappa )$
.
In particular, due to (*) above, 
holds for each
$j<\kappa $
, that is,
$J^{\prime }_i$
is independent over
$(A, J_{<i})$
. Thus by Fact 1.9(1) with an
$AJ_{<i}$
-automorphism, there is
$J^*_i=(d_{ij}\mid j<\kappa )$
such that
$J^*_i\equiv _{J_{<i}A} J^{\prime }_i$
(so that
$J_{<i}J^*_i\equiv _A(c_{mj}\mid m\in i,\ j\in \omega ) (c_{ij}\mid j\in \kappa )$
as well (**)), while 
(
) holds for each
$j<\kappa $
.
Finally, due to our choice of
$\kappa $
, we can apply Fact 1.2 to produce a desired
$L=(c^{\prime }_{ij}\mid j<\omega ),$
which is finitely based on
$J^*_i$
over
$J_{<i}B$
. Then due to (**) and strong indiscernibility over A, clearly
$J_i:=J_{<i}L $
satisfies (i). Moreover L is B-indiscernible, thus by (
) and the induction hypothesis for
$J_{<i}$
, (ii) for i holds too.
We put
$J:=\bigcup _{i<\mu } J_i=((c^{\prime }_{ij}\mid j\in \omega )\mid i\in \mu )$
as in Claim 2.1.1. Owing to the choice of
$\mu $
, we can use Fact 1.2 to find B-indiscernible
$I' =((c^{\prime \prime }_{ij}\mid j\in \omega )\mid i\in \omega )$
finitely based on J over B. As wanted,
$I'\equiv _A I$
because of strong indiscernibility. Now by left transitivity and base monotonicity,
$I'$
is a weak B-Morley array.
We begin to recall the notions of trees and tree indiscernibility from [Reference Dobrowolski, Kim and Ramsey3, Reference Kaplan and Ramsey4, Reference Kim, Kim and Scow8]. In this article, by a tree we mean a set
${\mathcal T}$
with a partial order
$\unlhd $
such that for every
$x\in {\mathcal T}$
, the set
$\{y \mid y \unlhd x\}\subseteq {\mathcal T}$
is linearly ordered by
$\unlhd $
. For any
$x,y\in {\mathcal T}$
, if there is a
$\unlhd $
-greatest element
$z \unlhd x,y$
, then we call z the meet of x and y, and write it as
$x\wedge y$
. Trees in this article also possess suitable notions of level and
$<_{\operatorname {lex}}$
. For example for ordinals
$\alpha , \gamma $
, we can see
$\gamma ^{\leq \alpha }$
as a tree in the language
$L_{s,\alpha }:=\{ \unlhd , \wedge ,<_{\operatorname {lex}}, (P_{\beta })_{\beta \leq \alpha } \}$
with the usual interpretations of
$\unlhd , \wedge ,<_{\operatorname {lex}}$
, and
$P_{\beta }=\{\eta \in \gamma ^{\leq \alpha }\mid \operatorname {dom}(\eta )=\beta \} $
, the set of elements with level
$\beta $
. In the rest, all the
$L_{s,\alpha }$
-structures we will consider are only trees.
Definition 2.2. Suppose that an
$L_{s,\alpha }$
-structure I is given.
-
(1) We say
$(a_{i} : i \in I)$
from
${\mathcal M}$
is s-indiscernible over A if whenever
$(s_{0}, \ldots , s_{n})$
,
$(t_{0}, \ldots , t_{n})$
are tuples from I with then we have
$$ \begin{align*}\text{qftp}_{L_{s,\alpha}}(s_{0}, \ldots, s_{n}) = \text{qftp}_{L_{s,\alpha}}(t_{0}, \ldots, t_{n}), \end{align*} $$
$a_{s_{0}}\ldots a_{s_{n}}\equiv _A a_{t_{0}}\ldots a_{t_{n}}.$
-
(2) We say
$(b_i\mid i\in I)$
is locally based on
$(a_{i} : i \in I)$
over A if given any finite set of formulas
$\Delta $
from
${\mathcal L}(A)$
and a finite tuple
$(t_{0}, \ldots , t_{n})$
from I, there is a tuple
$(s_{0}, \ldots , s_{n})$
from I such that and
$$\begin{align*}\text{qftp}_{L_{s,\alpha}} (t_{0}, \ldots, t_{n}) =\text{qftp}_{L_{s,\alpha}}(s_{0}, \ldots , s_{n}) \end{align*}$$
$$\begin{align*}\text{tp}_{\Delta}(b_{t_{0}}\ldots b_{t_{n}}) = \text{tp}_{\Delta}(a_{s_{0}} \ldots a_{s_{n}}).\end{align*}$$
-
(3) We say I-indexed trees have the modeling property if given any tree
$(a_{i} : i \in I)$
indexed by I and any set A there is s-indiscernible
$(b_{i} : i \in I)$
over A locally based on
$(a_{i} : i \in I)$
over A.
Fact 2.3. [Reference Kim, Kim and Scow8, Theorem 4.3] In any T, every
$\omega ^{<\omega }$
-indexed tree has the modeling property, where
$\omega ^{<\omega }$
is considered as an
$L_{s,\omega }$
-structure.
We now recall from [Reference Dobrowolski, Kim and Ramsey3, Reference Kaplan and Ramsey4] an
$L_{s,\alpha }$
-tree
$\mathcal {T}_{\alpha }$
, while making one change as suggested in [Reference Kim7]. Namely we consider all
$\beta \leq \alpha $
in the following definition (1) (so that every
${\mathcal T}_\alpha $
tree has a root), in order to make induction steps more palatable when we produce such trees of parameters possessing desired properties.
Definition 2.4. Suppose
$\alpha $
is an ordinal. We define
$\mathcal {T}_{\alpha }$
to be the set of functions
$\eta $
such that:
-
(1)
$\text {dom}(\eta )$
is an end-segment of
$\alpha $
of the form
$[\beta ,\alpha )$
for
$\beta \leq \alpha $
; -
(2)
$\text {ran}(\eta ) \subseteq \omega $
; -
(3) (finite support) the set
$\{\gamma \in \text {dom}(\eta ) : \eta (\gamma ) \neq 0\}$
is finite.
We interpret
$\mathcal {T}_{\alpha }$
as an
$L_{s,\alpha }$
-structure as follows:
-
• We put
$\eta \unlhd \xi $
if
$\eta \subseteq \xi $
. Write
$\eta \perp \xi $
if
$\neg (\eta \unlhd \xi )$
and
$\neg (\xi \unlhd \eta )$
. -
•
$\eta \wedge \xi := \eta \lceil {[\beta , \alpha )} = \xi \lceil {[\beta , \alpha ),}$
where (by finite support,
$$ \begin{align*}\beta = \text{min}\{ \gamma \leq \alpha : \eta\lceil {[\gamma, \alpha)} =\xi\lceil {[\gamma, \alpha)}\}\end{align*} $$
$\beta $
is not a limit ordinal).
-
• Put
$\eta <_{\operatorname {lex}} \xi $
if
$\eta \vartriangleleft \xi $
or,
$\eta \perp \xi $
with
$\text {dom}(\eta \wedge \xi ) = [\gamma +1,\alpha )$
and
$\eta (\gamma ) < \xi (\gamma )$
. -
• For
$\beta \leq \alpha $
and
$\eta :[\beta ,\alpha )\to \omega $
in
${\mathcal T}_\alpha $
, we write
$|\eta |:=\beta $
, the level of
$\eta $
. Then we put
$P_{\beta } := \{ \eta \in \mathcal {T}_{\alpha } : |\eta | = \beta \}$
. Note that
$P_{0}$
is the set of top level elements, and
$P_{\alpha }=\{\emptyset \}$
is the set of the root.
Definition 2.5. Suppose
$\alpha ,\beta $
are ordinals.
-
(1) (Concatenation) Suppose that
$\eta :[\beta ,\alpha )\to \omega $
in
${\mathcal T}_\alpha $
is given where
$\beta \leq \alpha $
. For
$i < \omega $
, we let
$\eta ^\smallfrown \langle i \rangle $
denote the function
$\eta \cup \{(\gamma ,i)\}: [\gamma ,\alpha )\to \omega $
in
${\mathcal T}_\alpha $
, if
$\beta =\gamma +1$
. In particular,
$\langle i\rangle :=\emptyset ^\smallfrown \langle i\rangle $
is in
${\mathcal T}_{\alpha }$
for any successor
$\alpha $
.On the other hand we let
$\langle i \rangle ^\smallfrown \eta $
denote the function
$\eta \cup \{(\alpha ,i)\}:[\beta ,\alpha +1)\to \omega $
, which is (due to our definition, always) in
${\mathcal T}_{\alpha +1}$
. -
(2) (Canonical inclusions) If
$\beta \leq \alpha $
, we define the map
$\iota _{\beta \alpha } : \mathcal {T}_{\beta } \to \mathcal {T}_{\alpha }$
by
$\iota _{\beta \alpha }(\eta ) := \eta \cup \{(\gamma , 0) : \gamma \in [\beta ,\alpha )\}$
. Note that due to finite support, if
$\alpha $
is a limit ordinal and
$\xi (\ne \emptyset )\in {\mathcal T}_\alpha $
, then there is
$\beta <\alpha $
and
$\eta \in {\mathcal T}_\beta $
such that
$\iota _{\beta \alpha }(\eta )=\xi $
, and hence
${\mathcal T}_\alpha =\bigcup _{\beta <\alpha } \iota _{\beta \alpha }({\mathcal T}_\beta )\cup \{\emptyset \}$
. -
(3) (The all
$0$
’s path) Given
$\beta \leq \alpha $
,
$\zeta _{\beta }\in {\mathcal T}_\alpha $
denotes the function with
$\text {dom}(\zeta _{\beta }) = [\beta , \alpha )$
and
$\zeta _{\beta }(\gamma ) = 0$
for all
$\gamma \in [\beta ,\alpha )$
.
We recall that due to compactness applied to Fact 2.3 we have the following modeling theorem for
${\mathcal T}_\alpha $
trees.
Fact 2.6. (T is any theory.) For any ordinal
$\alpha $
, all
${\mathcal T}_\alpha $
-indexed trees have the modeling property.
In the rest we assume trees of parameters are indexed by
$L_{s,\alpha }$
-trees having the modeling property (which we may simply say ‘modeling’ or ‘s-modeling’).
Definition 2.7. A sequence
$(A_i\mid i<\kappa )$
of trees of parameters is called mutually s-indiscernible over B if for each
$i<\kappa $
,
$A_i$
is s-indiscernible over
$BA_{\ne i}$
.
We will freely use the following remark throughout the article.
Remark 2.8.
-
(1) Let C be a tree of parameters. If C has the unique root
$c_\emptyset $
then C is s-indiscernible over A if and only if it is s-indiscernible over
$Ac_\emptyset $
. Now for any set B, due to s-modeling there is
$C'$
which is s-indiscernible over B locally based on C over B. If already C is s-indiscernible over
$B'(\subseteq B)$
then it easily follows that
$C\equiv _{B'} C'$
. -
(2) If a sequence
$(A_i\mid i<\omega )$
of trees is B-indiscernible and mutually s-indiscernible over B, then it is s-indiscernible over B. -
(3) For sets
$A,B$
and a tree of parameters
$C=(c_\eta \mid \eta \in I)$
, suppose that and C is s-indiscernible over B. If
$C'=(c^{\prime }_\eta \mid \eta \in I)$
is s-indiscernible over
$AB$
and locally based on C over
$AB$
, then still holds: If not, then there is a forking formula
$\varphi (x,b\bar c^{\prime }_{\bar \eta })$
with finite
$b\in B$
and a finite tuple
$\bar c^{\prime }_{\bar \eta }\in C'$
, realized by finite
$a\in A$
. Owing to local basedness, there is
$\bar c_{\bar \nu }\in C$
such that
$\varphi (a,b\bar c_{\bar \nu })$
holds, where
$\text {qftp}(\bar \eta )\equiv _{L_{s,\alpha }} \text {qftp}(\bar \nu )$
. Since
$C'$
is s-indiscernible over
$AB$
, we have
$\bar c^{\prime }_{\bar \eta }\equiv _{AB}\bar c^{\prime }_{\bar \nu }$
, in particular
$\models \varphi (a, b\bar c^{\prime }_{\bar \nu })$
as well, and
$\varphi (x,b\bar c^{\prime }_{\bar \nu })$
forks. Now due to (1), it follows
$C'\equiv _BC$
. Hence
$\varphi (x,b\bar c_{\bar \nu })$
is also a forking formula realized by a, which contradicts .
and C is s-indiscernible over B. If 
holds: If not, then there is a forking formula 
.The following lemma plays a key role in proving the main theorem. To show the lemma, we borrow techniques from [Reference Chernikov, Kim and Ramsey1] in order to produce s-indiscernible trees preserving suggested independence relations.
Lemma 2.9. Suppose that
$C_0{\kern-1pt}={\kern-1pt}(a_\eta \mid \eta {\kern-1pt}\in{\kern-1pt} {\mathcal T}_\alpha )$
is s-indiscernible and
${I{\kern-1pt}= {\kern-1pt}\langle b_i\mid i{\kern-1pt}<{\kern-1pt}\omega \rangle }$
is Morley, where
$b_0=a_\emptyset \in C_0$
. Then for any B, there is a sequence
$L=((a_{\eta ,i}\mid \eta \in {\mathcal T}_\alpha )\mid i<\omega )$
and
$B_0\equiv B$
such that:
-
(1)
$C_0=(a_{\eta ,0}\mid \eta \in {\mathcal T}_\alpha )$
; -
(2)
$\langle a_{\emptyset ,i}\mid i<\omega \rangle \equiv I$
and
$\langle a_{\emptyset ,i}\mid 0< i<\omega \rangle $
is Morley over
$(\emptyset , C_0B_0)$
; -
(3) L is
$B_0$
-indiscernible and mutually s-indiscernible over
$B_0$
(so s-indiscernible over
$B_0$
as well).
Proof. We first choose
$\kappa $
large enough with respect to
$|B|+|C_0|$
, and extend I to
$I_\kappa =\langle b_i\mid i<\kappa \rangle ,$
which is still Morley.
Claim 2.9.1. There are
$B_0\equiv B$
and
$J=(d_i\mid i<\kappa )\equiv _{b_0} I_\kappa $
with
$b_0=d_0$
such that:
-
(i) J is Morley over
$(\emptyset , B_0)$
; -
(ii)
$C_0$
is s-indiscernible over
$B_0J$
; -
(iii)
$J\smallsetminus \{b_0\}$
is independent over
$(\emptyset , C_0B_0)$
.
Proof of Claim 2.9.1.
By Fact 1.9(2), there is
$B^{\prime }_0\equiv B$
such that
$I_\kappa $
is Morley over
$(\emptyset , B^{\prime }_0)$
. Now by modeling, there is
$(a_\emptyset \in ) C^{\prime }_0$
such that
$C^{\prime }_0$
is s-indiscernible over
$a_\emptyset B^{\prime }_0$
locally based on
$C_0$
over
$a_\emptyset B^{\prime }_0$
.
Notice now that
$C_0\smallsetminus \{a_\emptyset \}$
is s-indiscernible over
$a_\emptyset $
. Thus by Remark 2.8(1) we have
$C_0\equiv _{a_\emptyset } C^{\prime }_0$
. Hence there is
$J'B_0$
such that
$J'B_0C_0\equiv _{a_\emptyset }I_\kappa B^{\prime }_0C^{\prime }_0$
. In particular
$C_0$
is s-indiscernible over
$B_0$
and
$J'$
is Morley over
$(\emptyset , B_0)$
(*). In order to further satisfy (ii) and (iii), we will find a suitable
$J\equiv _{B_0} J'$
as follows.
Write
$J':=(d^{\prime }_i\mid i<\kappa ),$
where
$d^{\prime }_0=b_0$
. Using induction on
$\kappa $
, we will construct
$J=(d_i\mid i<\kappa )$
(
$d_0=d^{\prime }_0$
) such that for each
$0<i<\kappa $
,
-
(a)
$d_{\leq i}\equiv _{d_0B_0} d^{\prime }_{\leq i}$
; -
(b)
$C_0$
is s-indiscernible over
$d_{\leq i}B_0$
; -
(c)
.
. Hence given
$0<i<\kappa $
, assume that we have obtained
$d_j$
for all
$(0<)j<i$
(as said
$d_0=d^{\prime }_0=b_0$
) satisfying (a)–(c). Now by extension with (a) and (*) above, there is
$d^*_i$
such that
$d_{<i}d^*_i\equiv _{d_0B_0} d^{\prime }_{\leq i}$
and 
(**). Then by modeling there is
$C^*_0$
s-indiscernible over
$d_{<i}d^*_iB_0$
locally based on
$C_0$
over
$d_{<i}d^*_iB_0$
(***). By the induction hypothesis for (b) (even for
$j=0$
) and Remark 2.8(1), we then have that
$ C^*_0\equiv _{d_{<i}B_0} C_0$
. Thus there is a wanted
$d_i$
such that
$d_{i} C_0\equiv _{d_{<i}B_0}d^*_iC^*_0$
. Therefore (a) and (b) for i hold. Lastly, due to Remark 2.8(3) applied to (**) and (***) with the induction hypothesis for (b), we have 
. Hence (c) for i also holds.
In conclusion we have produced J; and due to (a)–(c), together with
$B_0C_0$
, Claim 2.9.1 is verified.
We continue to work with
$J=(d_i\mid i<\kappa )$
and
$B_0$
produced in Claim 2.9.1. We further claim the following.
Claim 2.9.2. For each
$j<\kappa $
, there is
$C_j$
such that:
-
(i)*
$d_{\geq j}C_j\equiv _{B_0}JC_0$
; -
(ii)*
$\langle C_0,\dots ,C_j\rangle $
is mutually s-indiscernible over
$d_{>j}B_0$
.
Proof of Claim 2.9.2.
We suppose that for every
$j'<j$
we have obtained
$C_{j'}$
with the properties holding. If
$j=0$
then clearly
$C_0$
(with Claim 2.9.1(ii)) serves it.
Now consider the case
$0<j$
. Due to (ii)
$^*$
for all
$j'<j$
we have that
$(C_{j'}\mid j'<j)$
is mutually s-indiscernible over
$d_{\geq j}B_0$
(
$\star $
).
Moreover there is
$D^*_j $
such that
$d_{\geq j}D^*_j\equiv _{B_0} JC_0$
, since J is
$B_0$
-indiscernible (Claim 2.9.1(i)). Hence
$d_j$
is the root of the tree
$D^*_j$
, and
$D^*_j$
is s-indiscernible over
$d_{\geq j}B_0$
, by Claim 2.9.1(ii). Then by modeling we take
$D^{\prime }_j$
which is, over
$d_{\geq j}C_{< j}B_0 $
, s-indiscernible and locally based on
$D^*_j$
(
). By Remark 2.8(1) we have
$D^{\prime }_j\equiv _{d_{\geq j}B_0} D^*_j$
as well (
).
By now we choose
$ (D_{j'}\mid j'<j)$
as follows. If for all
$k< j'(<j)$
we have selected
$D_k$
, then by modeling we choose
$D_{j'}$
s-indiscernible over
$D_{<j'} C_{j'< <j} D^{\prime }_{j} d_{\geq j}B_0$
locally based on
$C_{j'}$
. We will show the following two properties:
-
(a)* The resulting
$\{ D_k \mid k< j\}\cup \{ D^{\prime }_j\} $
is mutually s-indiscernible over
$d_{> j}B_0$
. -
(b)*
$ D_{< j}\equiv _{ d_{\geq j}B_0} C_{< j}.$
For (a)
$^*$
, we firstly show that given
$k< j$
,
$D_k$
is s-indiscernible over
$D_{\ne k}D^{\prime }_jd_{> j}B_0$
. To lead a contradiction let us deny this. Hence there are
$$ \begin{align*}k_0<\dots <k_i=k<\dots <k_n< j \ (i\leq n);\end{align*} $$
$\varphi (\bar x_0, \dots , \bar x_n, \bar y)\in {\mathcal L}(d_{> j}B_0)$
; finite tuples of indices
$\bar \eta _0,\dots , \bar \eta _n, \bar \mu _i, \bar \nu \in {\mathcal T}_\alpha $
with
$$ \begin{align*}\text{qftp}(\bar \eta_i)\equiv_{L_{s,\alpha}} \text{qftp}(\bar \mu_i)\end{align*} $$
and corresponding tuples
$\bar u^{0}_{\bar \eta _{0}}\in D_{k_0},\dots , \bar u^{n}_{\bar \eta _{n}}\in D_{k_n}$
,
$\bar u^i_{\bar \mu _i }\in D_{k_i}$
,
$\bar w_{\bar \nu }\in D^{\prime }_{j}$
such that
$$ \begin{align*}\models \varphi(\bar u^0_{\bar \eta_0}, \dots, \bar u^i_{\bar \eta_i},\dots, \bar u^n_{\bar \eta_n}, \bar w_{\bar \nu})\wedge \neg \varphi(\bar u^0_{\bar \eta_0}, \dots;\bar u^i_{\bar \mu_i};\dots, \bar u^n_{\bar \eta_n}, \bar w_{\bar \nu}).\end{align*} $$
By the construction of
$D_{k_{i+1}},\dots , D_{k_n}$
and local basedness, we can consecutively find
$\bar v^n_{\bar \nu _n}\in C_{k_n}, \dots , \bar v^{i+1}_{\bar \nu _{i+1}}\in C_{k_{i+1}}$
(where of course
$ \text {qftp}(\bar \eta _n)\equiv _{L_{s,\alpha }} \text {qftp}(\bar v_n) $
, and so on) such that
$$ \begin{align*}\varphi(\bar u^0_{\bar \eta_0}, \dots, \bar u^i_{\bar \eta_i}; \bar v^{i+1}_{\bar \nu_{i+1}},\dots, \bar v^n_{\bar \nu_n}, \bar w_{\bar \nu})\wedge \neg \varphi(\bar u^0_{\bar \eta_0}, \dots; \bar u^i_{\bar \mu_i};\bar v^{i+1}_{\bar \nu_{i+1}},\dots, \bar v^n_{\bar \nu_n}, \bar w_{\bar \nu})\end{align*} $$
holds. But then due to the s-indiscernibility of
$D_k=D_{k_i}$
over the rest of the parameters, this leads to a contradiction.
Secondly, by the similar arguments with (
) above, it also easily follows that
$D^{\prime }_j$
is s-indiscernible over
$D_{< j}d_{> j}B_0$
. We have shown (a)
$^*$
.
Now (b)
$^*$
can be shown straightforwardly, by iterated applications of Remark 2.8(1) using (
$\star $
) above and the way of choosing
$D_{<j}$
.
We can now return our attention to finding wanted
$C_j$
. Due to (b)
$^*$
above, there is
$C_j$
such that
$$ \begin{align*}C_{< j} C_j \equiv_{ d_{\geq j} B_0} D_{< j}D^{\prime}_j.\end{align*} $$
Then due to above (
) and (a)
$^*$
, it follows that
$C_j$
and
$C_{\leq j}$
satisfy (i)
$^*$
and (ii)
$^*$
for j. We have shown Claim 2.9.2.
We are about to finish our proof of Lemma 2.9. We work again with the selected
$J=(d_i\mid i<\kappa )$
and
$B_0$
from Claim 2.9.1, and
$\langle C_i\mid i<\kappa \rangle $
from Claim 2.9.2. In particular,
$d_{\geq i}C_i\equiv _{B_0}JC_0$
and
$(C_i\mid i<\kappa )$
is mutually s-indiscernible over
$B_0$
(
$\sharp $
). Now since
$\kappa $
is sufficiently large, by Fact 1.2, there is
$B_0$
-indiscernible
$(C^{\prime }_i\mid i<\omega )$
finitely based on
$(C_i\mid i<\kappa )$
over
$B_0$
. By an automorphism we can assume
$C^{\prime }_0=C_0$
. Then we put a wanted L as
$(C^{\prime }_i\mid i<\omega )$
. Clearly (1) holds. Since
$J\equiv I_\kappa $
(Claim 2.9.1), the first condition of (2) is satisfied. The second condition of (2) holds as well, due to that
$(C^{\prime }_i\mid 0<i<\omega )$
is
$B_0C_0$
-indiscernible, and Claim 2.9.1(iii) with (
$\sharp $
). Moreover (3) comes from (
$\sharp $
). The proof of Lemma 2.9 is completed.
3. Existence and the main theorem
In this section we consider the case where Kim-forking fails to have existence. Hence after naming parameters, we have consistent formulas
$\psi (x)\in {\mathcal L}$
and
$\varphi _i(x,b_i)$
(
$i\leq n$
) such that
$\models \psi (x)\rightarrow \varphi _0(x,b_0)\vee \dots \vee \varphi _n(x,b_n)$
, and each
$\varphi _i(x,b_i)$
Kim-divides. Then replacing
$\varphi _i(x,b_i)$
by
$\psi (x)\wedge \varphi _i(x,b_i)$
(and renaming it as
$\varphi _i(x,b_i)$
) allow us the following.
In this section we assume
$$ \begin{align*}\models \psi(x)\leftrightarrow \varphi_0(x,b_0)\vee\dots \vee \varphi_n(x,b_n)\end{align*} $$
holds (all are consistent formulas) and each
$\varphi _i(x,b_i)$
(
$i\leq n$
) Kim-divides witnessed by a Morley sequence
$I_i=(b_i^j\mid j<\omega ),$
where
$b_i^0=b_i$
. We now put
$D:=b_0\dots b_n$
. Notice that for any
$b^{\prime }_i\equiv b_i$
,
$\varphi _i(x,b^{\prime }_i)\models \psi (x)$
, and for any
$b^{\prime }_0\dots b^{\prime }_n\equiv D$
,
$ \psi (x)$
is equivalent to
$\varphi _0(x,b^{\prime }_0)\vee \dots \vee \varphi _n(x,b^{\prime }_n)$
as well.
Lemma 3.1. Given a cardinal
$\kappa $
and each ordinal
$\alpha <\kappa $
, there is s-indiscernible
$C_\alpha =(a_\eta \mid \eta \in {\mathcal T}_\alpha )$
and
$D_\alpha =b_0^\alpha \dots b_n^\alpha \equiv D$
, and there is
$f: \kappa \to n+1$
, which satisfy the following:
-
(1) Suppose that
$\alpha =\gamma +1$
. Then-
(a)
$(a_{\langle i\rangle }\mid i\in \omega )\equiv I_{f(\gamma )}$
(where
$\langle i\rangle \in {\mathcal T}_\alpha $
); -
(b) for each
$0<i<\omega $
, we have that
-
-
(2)
$C_\alpha $
is s-indiscernible over
$D_\alpha $
, and
$a_\emptyset =b^\alpha _{f(\alpha )}\in D_\alpha $
. -
(3) For every
$\eta \in P_0$
,
$\{\varphi _{f(|\nu |)}(x, a_{\nu })\mid \nu \unlhd \eta \}$
is consistent. -
(4) For
$\alpha '<\alpha $
,
$C_{\alpha '}\subseteq C_\alpha $
via the canonical inclusion.
Proof. Suppose that we have determined
$C_\beta , D_\beta $
, and
$f(\beta ) $
for all
$\beta <\alpha $
satisfying the properties. We want to choose
$C_\alpha ,D_\alpha $
, and
$f(\alpha )$
.
If
$\alpha =0$
then choose
$C_0$
to be any
$b_i\in D_0:=D$
and
$f(0)=i$
. This choice clearly works.
Now if
$\alpha $
is a limit ordinal, notice that
$C_{<\alpha }$
, the union of
$C_\beta $
for all
$\beta <\alpha $
is also s-indiscernible, due to the canonical inclusion. Then by modeling we choose
$C^{\prime }_{<\alpha }$
, which is s-indiscernible over D and locally based on
$C_{<\alpha }$
over D. Then since
$C_{<\alpha }$
is s-indiscernible, we have
$C_{<\alpha } \equiv C^{\prime }_{<\alpha }$
. Hence there is a wanted
$D_\alpha =b_0^\alpha \dots b_n^\alpha $
such that
$C_{<\alpha } D_\alpha \equiv C^{\prime }_{<\alpha }D$
, so
$C_{<\alpha }$
is s-indiscernible over
$D_\alpha $
(*). Notice that by the induction hypothesis with finite support of
${\mathcal T}_{<\alpha }$
, given
$\eta \in P_0$
(of
${\mathcal T}_{<\alpha }$
),
$\{\varphi _{f(|\nu |)}(x, a_{\nu })\mid \nu \unlhd \eta \}$
is also consistent. Moreover due to s-indiscernibility, for
$\nu \in P_\beta $
, we have
$a_\nu \equiv b_{f(\beta )}$
. Thus obviously
$$ \begin{align*}\{\varphi_{f(|\nu|)}(x, a_{\nu})\mid \nu\unlhd \eta\}\models \psi(x).\end{align*} $$
Since
$\models \psi (x)\leftrightarrow \varphi _0(x,b^\alpha _0)\vee \dots \vee \varphi _n(x,b^\alpha _n)$
as well, there is some
$i_0\leq n$
such that
$$ \begin{align*}\{\varphi_{f(|\nu|)}(x, a_{\nu})\mid \nu\unlhd \eta\}\cup \{\varphi_{i_0}(x,b^\alpha_{i_0})\}\end{align*} $$
is consistent (**). Then we put
$f(\alpha ):=i_0$
and
$a_\emptyset :=b^\alpha _{i_0}$
(in
$C_\alpha $
indexed by
${\mathcal T}_\alpha $
). Since
$a_\emptyset \in D_\alpha $
,
$C_\alpha $
is s-indiscernible over
$D_\alpha $
, due to (*) above. The rest of (2) and (4) for
$\alpha $
are already satisfied. Moreover since
$C_\alpha $
is s-indiscernible over its root
$b^\alpha _{i_0}$
, (3) for
$C_\alpha $
also holds by (**).
It remains to check when
$\alpha =\gamma +1$
. By the induction hypothesis,
$C_\gamma $
is s-indiscernible. Then by applying Lemma 2.9 to
$C_\gamma $
and Morley
$I_{f(\gamma )}$
, we find a sequence
$L=((a_{\eta ,i}\mid \eta \in {\mathcal T}_\alpha )\mid i<\omega )$
with
$D_\alpha \equiv D$
such that:
-
(i)
$C_\gamma =(a_{\eta ,0}\mid \eta \in {\mathcal T}_\alpha )$
; -
(ii)
$\langle a_{\emptyset ,i}\mid i<\omega \rangle \equiv I_{f(\gamma )}$
and
$\langle a_{\emptyset ,i}\mid 0< i<\omega \rangle $
is Morley over
$(\emptyset , C_\gamma D_\alpha )$
; -
(iii) L is
$D_\alpha $
-indiscernible and mutually s-indiscernible over
$D_\alpha $
(so s-indiscernible over
$D_\alpha $
as well).
In order to find a wanted tree
$C_\alpha =(a_\eta \mid \eta \in {\mathcal T}_\alpha )$
, we canonically put
$a_{\rhd \emptyset }:=L$
. Then (1) for
$\alpha $
is fulfilled due to (ii) above. Now using the same argument in the previous paragraph, we can find
$j_0\leq n$
so that we put
$f(\alpha ):=j_0$
;
$a_\emptyset :=b^\alpha _{j_0}\in D_\alpha $
; and (2)–(4) for
$\alpha $
again follow due to (iii).
We are ready to show the main theorem.
Theorem 3.2. Assume T is NSOP
${}_1$
. Then Kim-forking satisfies existence. Equivalently any complete type has an 
-Morley sequence in it.
Proof. We keep our starting assumption in this section that Kim-forking fails to have existence, and work with chosen
$b_0\dots b_n$
and
$I_i$
. We will deduce a contradiction under T being NSOP
${}_1$
(which is preserved regardless of naming parameters).
We also work with Lemma 3.1 and selected
$C_\alpha $
. There we take
$\kappa $
sufficiently large (to apply Fact 1.2) and in the tree
$C_{<\kappa }=\bigcup _{\alpha <\kappa } C_\alpha =(a_\eta \mid \eta \in {\mathcal T}_{<\kappa }) $
, we consider the sequence
$\langle L_\alpha \mid \alpha <\kappa \rangle $
, where
$L_\alpha =(a_{{\zeta _{\alpha +1}}^\smallfrown \langle i\rangle }\mid i\in \omega ). $
By Fact 1.2, there is an indiscernible sequence
$$ \begin{align*}J_{<\omega}:=\langle J_i=(c_{i,j}\mid j<\omega) \mid i<\omega\rangle\end{align*} $$
finitely based on
$\langle L_\alpha \mid \alpha <\kappa \rangle $
. Notice that due to Lemma 3.1, we have that all
$c_{ij}$
and some
$b_i$
, say
$b_0$
have the same type; and
$\{\varphi _0(x, c_{i,0})\mid i<\omega \}$
is consistent. Moreover due to s-indiscernibility, each
$J_i$
is
$J_{>i}$
-indiscernible. Then since T is NSOP
${}_1$
, it follows that
$\{\varphi _0(x, c_{i,1})\mid i<\omega \}$
is consistent as well, by Fact 1.7. Again due to s-indiscernibility from Lemma 3.1, we more have that
$$ \begin{align*}J^{\prime}_{<\omega}:=\langle J^{\prime}_i=(c_{i,j}\mid 1\leq j<\omega) \mid i<\omega\rangle\end{align*} $$
is a Morley array, in particular strongly indiscernible.
Now there is a model M such that
$J^{\prime }_{<\omega }$
is weak Morley over M, due to Lemma 2.1. Hence both
$(c_{i,1}\mid i<\omega )$
and
$(c_{0,j}\mid 1\leq j<\omega )$
are Morley in
$\operatorname {tp}(c_{0,1}/M)$
. However since
$(c_{0,j}\mid 1\leq j<\omega )\equiv I_0$
as in Lemma 3.1,
$\{\varphi _0(x, c_{0,j})\mid 1\leq j<\omega \}$
is inconsistent. This contradicts Kim’s lemma over a model in NSOP
${}_1 T$
. We have proved the theorem.
We give two remarks before finishing this section.
Remark 3.3.
-
(1) If an arbitrary theory T satisfies existence, then existence for
follows trivially since implies . On the other hand, if in T there is no Morley sequence over
$\emptyset $
at all, then existence for over
$\emptyset $
again follows trivially since there is no Kim-dividing formula over
$\emptyset $
.However even if existence fails to hold in an NSOP
${}_1 T$
so that say
$p(x)\in S(\emptyset )$
forks over
$\emptyset $
, there may exist Morely sequences in other complete types over
$\emptyset $
. Thus even in the case, the proof of Theorem 3.2 is not trivialized. For example, as mentioned in the beginning, Mutchnik in [Reference Mutchnik9] gives a two-sorted example of NSOP
${}_1$
theory without existence. In one sort, Morley sequences (over
$\emptyset $
) do not exist, while they exist in the other sort. -
(2) Originally, in [Reference Kaplan and Ramsey4, Definition 3.13] Kaplan and Ramsey define
using invariant Morley sequences, which, as proved in [Reference Kaplan and Ramsey4, Theorem 7.7], coincide with Definition 1.4(1) and (2) using Morley sequences when the theory is NSOP
${}_1$
and the base is a model. In any theory, existence for (as defined in Definition 1.4(1) and (2)) implies existsnce for defined using invariant Morley sequences, since every invariant Morley sequence is a Morley sequence. Recently, in an unpublished note, James Hanson shows with respect to invariant Morley sequences satisfies existence in any theory.
follows trivially since 
implies 
. On the other hand, if in T there is no Morley sequence over 
over 
using invariant Morley sequences, which, as proved in [
(as defined in Definition 
defined using invariant Morley sequences, since every invariant Morley sequence is a Morley sequence. Recently, in an unpublished note, James Hanson shows 
with respect to invariant Morley sequences satisfies existence in any theory.4. Discussion
In this section we will discuss a couple of issues. First of all it is natural to ask whether there is a theory T where Kim-forking does not have existence. As proved in Section 3, if exists then it should not be NSOP
${}_1$
.
As pointed out in Section 1, after Fact 1.3, the typical circle example with the ternary R does not have existence (over
$\emptyset $
). It is because there are three distinct points
$a_0,a_1,a_2$
on the circle such that
$\models x=x\leftrightarrow R(a_0,x,a_1)\vee R(a_1,x,a_2) \vee R(a_2,x,a_0)$
, and each
$R(a_i,x,a_{i+1})$
(mod
$3$
) divides over
$\emptyset $
. But the dividing of each formula is witnessed not by a Morley sequence but only by an 
-Morley sequence. Thus this (although not NSOP
${}_1$
) does not serve as an example where Kim-forking fails to have existence.
With respect to this question, at least we observe the following.
Remark 4.1.
-
(1) Suppose that a consistent formula
$\psi (x)\in {\mathcal L}$
forks, witnessed by two consistent formulas
$ \varphi (x, a_0)$
and
$\phi (x,b_0)$
. Namely
$\models \psi (x)\leftrightarrow \varphi (x,a_0)\vee \phi (x,b_0)$
and dividing of
$\varphi (x,a_0)$
and
$\phi (x,b_0)$
is witnessed by indiscernibles
$I=(a_i\mid i<\omega )$
and
$J=(b_i\mid i<\omega )$
, respectively. We claim that I (hence J as well) can neither be -Morley, nor Morley. (In general, the reversed sequence of a non (-)Morley sequence can be (-)Morley. But the proof below says I is not -Morley in either direction. Moreover the circle example above says the argument below does not work if three (or more) formulas witness forking of
$\psi (x)$
.)Note that there are
$ k(\geq 1)$
and
$k'\geq 2$
such that
$\varphi (x, a_0)\ (k+1)$
-divides, but not k-divides, via I; and
$\phi (x,b_0)\ k'$
-divides, via J (*). Consider an
${\mathcal L}$
-formula Clearly
$$ \begin{align*}\chi(x_k,\dots, x_0)&:= \exists z(\psi(z)\wedge \varphi(z,x_k)\wedge \dots \wedge\varphi(z,x_1))\wedge \neg\exists z' (\psi(z')\\& \quad \wedge \varphi(z',x_k)\wedge \dots \wedge\varphi(z',x_0)).\end{align*} $$
$(a_k,\dots , a_0)$
realizes
$\chi $
. It suffices, by left transitivity of , to show that
$\chi (x_k,\dots ,x_1;a_0)$
divides.
By compactness there clearly is an indiscernible
$(a^j_0b^j_0\mid j<\omega )$
such that
$ a^0_0b^0_0= a_0b_0$
, and
$(b_0^j\mid j\in \omega )\equiv J$
. Then
$\chi (x_k,\dots ,x_1;a_0)\ k'$
-divides via
$(a^j_0\mid j<\omega )$
, namely,
$\chi (x_k,\dots ,x_1;a^0_0)\wedge \dots \wedge \chi (x_k,\dots ,x_1;a^{k'-1}_0)$
is inconsistent. Why? Suppose not, say realized by
$c_k\dots c_1$
. Note that for each
$j<k'$
, Since
$$ \begin{align*}\psi(z)\wedge \varphi(z,c_k)\wedge \dots \wedge\varphi(z,c_1)\models \neg \varphi (z,a^j_0)\wedge \psi(z)\models \phi(z, b^j_0).\end{align*} $$
$ \psi (z)\wedge \varphi (z,c_k)\wedge \dots \wedge \varphi (z,c_1)$
is consistent,
$\{\phi (x,b^j_0)\mid j<k'\}$
is then consistent, contradicting (*) above.
-
(2) Consider a 2-sorted structure
$(\mathbb {N},\mathcal {P}(\mathbb {N}),\in ),$
where
$\in $
is the usual membership relation. One can find sequences
$(B_i)_{i<\omega }$
and
$(B^{\prime }_i)_{i<\omega }$
in
$\mathcal {P}(\mathbb {N})$
satisfying
$B^{\prime }_0=\mathbb {N}\setminus B_0$
,
$B_i\cap B_j=\emptyset $
,
$ B^{\prime }_i \cap B^{\prime }_j=\emptyset $
for each
$i\neq j$
, and
$|B_i|=|\mathbb {N}\setminus B_i|=|B^{\prime }_i|=|\mathbb {N}\setminus B^{\prime }_i|=\aleph _0$
for each i. Since any permutation of
$\mathbb {N}$
induces an automorphism of the structure, it easily follows that
$\operatorname {tp}(B_i)=\operatorname {tp}(B^{\prime }_j)$
for all
$i,j$
. Put
$B=B_0, B'=B^{\prime }_0$
. Now
$\models x=x\leftrightarrow x\in B\vee x\notin B$
. Moreover each of the formulas
$x\in B$
, and
$x\notin B$
(equivalent to
$x\in B'$
) 2-divides over
$\emptyset $
, witnessed by the sequences
$(B_i)_{i<\omega }$
and
$(B^{\prime }_i)_{i<\omega }$
, respectively. This is an example of (1), which is a variation of an example failing existence mentioned in [Reference Shelah11].
-Morley, nor Morley. (In general, the reversed sequence of a non (
-)Morley sequence can be (
-)Morley. But the proof below says I is not 
-Morley in either direction. Moreover the circle example above says the argument below does not work if three (or more) formulas witness forking of 
, to show that 
Now we list the following questions.
Question 4.2.
-
(1) Is there a theory where Kim-forking does not have existence? As mentioned in Remark 3.3(2), if Kim-forking is defined using invariant Morley sequences, then every theory has Kim-forking existence.
-
(2) In NSOP
${}_1 T$
, can one obtain the same Theorem 3.2 even if each
$I_i$
in Section 3 is weakened to be an -Morley sequence?
-Morley sequence? In addition to the questions above, in all NSOP
${}_1$
theories we are now able to define a new kind of dividing witnessed by 
-Morley sequences, due to the main result Theorem 3.2. As mentioned before Section 1, it is intriguing to know how far a theoretical development can be made using the new dividing for NSOP
${}_1$
theories, although the new one is known to be equal to Kim-dividing and Kim-forking in NSOP
${}_1$
theories with existence.
By the way, a quick consequence of Theorem 3.2 is as follows.
Remark 4.3. Assume T is NSOP
${}_1$
. Define 
if there is a model
$(B\subseteq ) M$
such that 
and 
.
-
• (Symmetry of
) Due to symmetry of over a model, iff . -
• (Existence of
) Given
$A, B$
, by Theorem 3.2, there is
$(B\subseteq )M$
such that . Then since
$B\subseteq M$
, we have . In other words, holds. -
• If T has existence then
iff [Reference Chernikov, Kim and Ramsey1, Proposition 3.5].
) Due to symmetry of 
over a model, 
iff 
.
) Given 
. Then since 
. In other words, 
holds.
iff 
[However it is unclear whether in T, 
has other properties, such as extension, transitivity, and so on.
Acknowledgments
The authors thank Nicholas Ramsey for his explanation about the argument in [Reference Chernikov, Kim and Ramsey1], which is critically used in the proof of Claim 2.9.2. The authors also thank Ikuo Yoneda for his valuable discussion, which helped us refine our article. The authors are grateful to James Hanson and Scott Mutchnik for helpful discussions about their results. The authors thank the anonymous referee for improving the example in Remark 4.1(2).
Funding
All authors are supported by an NRF of Korea grant 2021R1A2C1009639. The second author is supported by a KIAS individual grant, project no. 6G091801. The third author is supported by the 2023 and 2024 Yonsei University Post-Doctoral Researcher Supporting Program, project nos. 2023-12-0159 and 2024-12-0214, and the G-LAMP Program of the NRF of Korea grant funded by the Ministry of Education (no. RS-2024-00441954).
