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Nice Properties of Theories

ω-stable
A theory is ω-stable if SM
n
(A) is countable for all models M and countable AM.

See , , .
superstable
Given an infinite cardinal κ, a theory is κ-stable if
|SM
n
(A)| ≤ κ
for all models M and AM of size at most κ.

A theory T is superstable if there is some cardinal λ such that T is κ-stable for all κ ≥ λ.

See , , .

Remarks
In the definition, λ can be made equal to 2|T|.
stable
Given an infinite cardinal κ, a theory is κ-stable if
|SM
n
(A)| ≤ κ
for all models M and AM of size at most κ.

A theory is stable if it is κ-stable for some infinite κ.

See , , .

Alternate definition:
A formula $$\varphi(x,y)$$ has the order property if there are $$(a_i)_{i\lt\omega}$$ and $$(b_i)_{i\lt\omega}$$ such that $\models\varphi(a_i,b_j)~\Leftrightarrow~ i\lt j.$ A theory is stable if no formula has the order property.

See .
o-minimal
Assume the language $$\mathcal{L}$$ contains a binary relation <, and that T is an $$\mathcal{L}$$-theory in which < is a linear order. T is o-minimal if, given a model M, every definable subset of M can be written as a finite union of points in M and intervals with endpoints in M.

characterization of forking

See , , .
dp-minimal
A theory has dp-rankn if there are formulas $$\varphi_1(x,y),\ldots,\varphi_n(x,y)$$ and mutually indiscernible sequences $$(a^1_i)_{i\lt\omega},\ldots,(a^n_i)_{i\lt\omega}$$ such that for any function $$\sigma:\{1,\ldots,n\}\longrightarrow\omega$$, the type $\{\varphi_k(x,a^k_{\sigma(k)}):k\leq n\}\cup$ $\{\neg\varphi_k(x,a^k_i):i\neq\sigma(k),~k\leq n\}$ is consistent.

A theory is dp-minimal if it has dp-rank 1.

See , .
NIP (dependent)
A formula $$\varphi(x,y)$$ as the independence property (IP) if there are $$(a_i)_{i\lt\omega}$$ and $$(b_I)_{I\subseteq\omega}$$ such that $\models\varphi(a_i,b_I)~\Leftrightarrow~i\in I.$ A theory is NIP (dependent) if no formula has the independence property.

See , .
supersimple
A theory is supersimple if for all sets B and complete types p in Sn(B), there is a finite AB such that p does not fork over A.

See , .
simple
A theory T is simple if for all sets B and complete types p in Sn(B) there is AB such that |A|≤|T| and p does not fork over A.

Alternate Definition:
A formula $$\varphi(x,y)$$ has the tree property (TP) if there are $$(a_\eta:\eta~\in\omega^{\lt\omega})$$ and some $$k\geq 2$$ such that
• $$\forall~\sigma\in\omega^\omega$$, $$\{\varphi(x,a_{\sigma|_n}):n\lt\omega\}$$ is consistent.
• $$\forall~\eta\in\omega^{\lt\omega}$$, $$\{\varphi(x,a_{\sigma\hat{~}n}:n\lt\omega\}$$ is $$k$$-inconsistent.
A theory is simple (NTP) if no formula has the tree property.

See , .

Remarks:
One often studies theories in which forking independence is well-behaved. Simple theories have been shown to strongly exemplify this. In particular, the following are equivalent (see , ):
• T is simple.
• Nonforking (or nondividing) is symmetric.
• Nonforking (or nondividing) is transitive.
• Nonforking (or nondividing) has local character.
• There is a notion of independence (which must be nonforking) satisfying certain properties (see ).
NSOP1
A formula $$\varphi(x,y)$$ has SOP1 if there are $$(a_\eta:\eta~\in 2^{\lt\omega})$$ such that
• $$\forall~\sigma\in 2^\omega$$, $$\{\varphi(x,a_{\sigma|_n}):n\lt\omega\}$$ is consistent.
• $$\forall~\eta,\mu\in 2^{\lt\omega}$$, if $$\mu\hat{~}0\prec\eta$$, then $$\{\varphi(x,a_{\mu\hat{~}1}),~\varphi(x,a_{\eta})\}$$ is inconsistent.
A theory is NSOP1 if no formula has SOP1.

See , , .
NTP1 (NSOP2)
no proper examples known
A formula $$\varphi(x,y)$$ has the tree property 1 (TP1) if there are $$(a_\eta:\eta~\in \omega^{\lt\omega})$$ such that
• $$\forall~\sigma\in \omega^\omega$$, $$\{\varphi(x,a_{\sigma|_n}):n\lt\omega\}$$ is consistent.
• $$\forall$$ incomparable $$\eta,\mu\in\omega^{\lt\omega}$$, $$\{\varphi(x,a_\mu),~\varphi(x,a_{\eta})\}$$ is inconsistent.
A theory is NTP1 if no formula has TP1.

See , .

Remarks
As a property of the theory, TP1 is equivalent to SOP2 (see ), which was recently shown to be maximal in Keisler's order (see ).
NSOP3
no proper examples known
For $$n\geq 3$$, a formula $$\varphi(x,y)$$ has the n-strong order property (SOPn) if there are $$(a_i)_{i\lt\omega}$$ such that $$\models\varphi(a_i,a_j)$$ for all $$i \lt j$$; but $\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}$ is inconsistent.

A theory is NSOPn if no formula has SOPn.

See , .
NSOP4
For $$n\geq 3$$, a formula $$\varphi(x,y)$$ has the n-strong order property (SOPn) if there are $$(a_i)_{i\lt\omega}$$ such that $$\models\varphi(a_i,a_j)$$ for all $$i\lt j$$; but $\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}$ is inconsistent.

A theory is NSOPn if no formula has SOPn.

See , , .

Remarks:
There are results about non-existence of universal models of cardinality λ (for certain λ) when T has SOP4 (see , ).
For example:
If T has SOP4 and λ is regular such that κ+<λ<2κ for some κ, then T has no universal model of cardinality λ.
NSOPn+1
For $$n\geq 3$$, a formula $$\varphi(x,y)$$ has the n-strong order property (SOPn) if there are $$(a_i)_{i\lt\omega}$$ such that $$\models\varphi(a_i,a_j)$$ for all $$i\lt j$$; but $\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}$ is inconsistent.

A theory is NSOPn if no formula has SOPn.

See , .
NFSOP
A type $$p(x,y)$$, with $$|x|=|y|$$, has the strong order property if there are $$(a_i)_{i\in\omega}$$ such that $$p(a_i,a_j)$$ holds for all $$i\lt j$$, but $p(x_1,x_2)\cup \ldots\cup p(x_{n-1},x_n)\cup p(x_n,x_1)$ is unsatisfiable for all $$n\ge 2$$.

A theory is NFSOP (no finitary strong order property) if no type $$p(x,y)$$ has the strong order property with $$|x|$$ finite.

See , .

Remarks:
Two other properties of theories with a close relationship to the finitary strong order property are the fully finitary strong order property and the strong order property. The definitions of these properties are obtained by replacing the type in the definition above with, respectively, a formula or a type in arbitrarily many variables. In particular, the strong order property for a theory is equivalent to SOPn for all n ≥ 3.
NSOP
A formula $$\varphi(x,y)$$ has the strict order property (SOP) if there are $$(a_i)_{i\lt\omega}$$ such that $\models\exists x(\varphi(x,a_j)\wedge\neg\varphi(x,a_i))~\Leftrightarrow~i\lt j.$ A theory is NSOP if no formula has the strict order property.

See , .

NTP2
A formula $$\varphi(x,y)$$ has the tree property 2 (TP2) if there are $$(a_{i,j})_{i,j\lt\omega}$$ such that
• $$\forall~\sigma\in\omega^\omega$$, $$\{\varphi(x,a_{n,\sigma(n)}):n\lt\omega\}$$ is consistent.
• $$\forall~n\lt\omega,~\forall~ i\lt j\lt\omega$$, $$\{\varphi(x,a_{n,i}),~\varphi(x,a_{n,j})\}$$ is inconsistent.
A theory is NTP2 if no formula has the tree property 2.

See , .

distal
A theory T is distal if, for any parameter set A, any A-indiscernible sequence I, and any tuple b, if I = I1 + I2, for some sequences I1 and I2 without endpoints, and I1+b+I2 is indiscernible, then I1+b+I2 is A-indiscernible.

See , .

Remarks:
The following are two useful characterizations of distal theories.
• A theory is distal if and only if all generically stable measures are smooth.

• (Strong honest definitions) An NIP theory T is distal if and only if for any formula $$\varphi(x,y)$$ there is a formula $$\psi(x,y_1,\ldots,y_n)$$, with $$|y_i|=|y|$$, such that for any finite set B of $$y$$-tuples, with |B|≥ 2, and any $$x$$-tuple $$a$$, there are $$b_1,\ldots,b_n$$ in B such that $$\psi(x,b_1,\ldots,b_n)\vdash\textrm{tp}_{\varphi}(a/B)$$.
nonforking exists
strongly minimal
A theory is strongly minimal if for all models M, any definable subset of M is finite or cofinite.

See , .

Remarks
For simplicity, this region of the map is for strongly minimal theories in a countable language. With this caveat, a strongly minimal theory is ω-stable, as indicated by the map.
Click a property above to highlight region and display details. Or click the map for specific region information.
List of Examples
Implications Between Properties
Click the link below to see the implications between all the above properties:

Diagram of Implications

Some other important equivalences:
• stable ⇔ NIP and NSOP
• simple ⇔ NTP1 and NTP2
For proofs of these implications, see , , , , , , .
Some of the proofs have been collected here.
Open Regions
Each small region outlined in red currently contains no known examples, due to the following open questions:
• Are the implications:
SOP3 ⇒ TP1 ⇒ SOP1
proper? See , , .
• Is there a theory which is NSOP and NTP2, but not simple? See , Exercise III.7.12.
Open Examples
This list is intended for popular and/or important examples of theories with unknown positions on the map. Suggestions for such theories are welcomed.