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Nice Properties of Theories

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

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

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

        See , , .

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

        See .
        o-minimal
        o-minimal
          Assume the language \(\large\mathcal{L}\) contains a binary relation <, and that T is an \(\large\mathcal{L}\)-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
          dp-minimal
            A theory has dp-rank ≥ n if there are formulas \(\large\varphi_1(x,y),\ldots,\varphi_n(x,y)\) and mutually indiscernible sequences \(\large(a^1_i)_{i\lt\omega},\ldots,(a^n_i)_{i\lt\omega}\) such that for any function \(\large\sigma:\{1,\ldots,n\}\to\omega\), the type \[\large\{\varphi_k(x,a^k_{\sigma(k)}):k\leq n\}~\cup\] \[\large\{\neg\varphi_k(x,a^k_i):i\neq\sigma(k),~k\leq n\}\] is consistent.

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

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

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

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

                  Alternate Definition:
                  A formula \(\large\varphi(x,y)\) has the tree property (TP) if there are \(y\)-tuples \(\large(a_\eta)_{\eta~\in\omega^{\lt\omega}}\) and some \(\large k\geq 2\) such that
                  • \(\large \{\varphi(x,a_{\sigma|n}):n\lt\omega\}\) is consistent for all \(\large \sigma\in\omega^\omega\),
                  • \(\large \{\varphi(x,a_{\eta\hat{~}i}:i\lt\omega\}\) is
                    \(\large k\)-inconsistent for all \(\large \eta\in\omega^{\lt\omega}\).
                  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 ).
                  NTP1 (NSOP1/NSOP2)
                  NTP1 (NSOP1/NSOP2)
                    Ad hoc terminology:
                    Given a formula \(\large\varphi(x,y)\) and an ordinal \(\large \alpha\), an \(\large \alpha\)-tree for \(\large \varphi(x,y)\) is a sequence \(\large (b_\eta)_{\eta~\in \alpha^{\lt\omega}}\) of \(\large y\)-tuples such that for all \(\large \sigma\in \alpha^\omega\), \(\large \{\varphi(x,b_{\sigma|n}):n\lt\omega\}\) is consistent.

                    \(\large \varphi(x,y)\) has the tree property of the first kind (TP1) if there is an \(\large \omega\)-tree \(\large (b_\eta)_{\eta~\in \omega^{\lt\omega}}\) for \(\large \varphi(x,y)\) such that for all incomparable \(\large \eta,\mu\in \omega^{\lt\omega}\), \(\large \{\varphi(x,b_\eta),~\varphi(x,b_\mu)\}\) is inconsistent.

                    \(\large \varphi(x,y)\) has SOP2 if there is a 2-tree \(\large (b_\eta)_{\eta~\in 2^{\lt\omega}}\) for \(\large \varphi(x,y)\) such that for all incomparable \(\large \eta,\mu\in 2^{\lt\omega}\), \(\large \{\varphi(x,b_\eta),~\varphi(x,b_\mu)\}\) is inconsistent.

                    \(\large \varphi(x,y)\) has SOP1 if there is a 2-tree \(\large (b_\eta)_{\eta~\in 2^{\lt\omega}}\) for \(\large \varphi(x,y)\) such that for all \(\large \eta,\mu\in 2^{\lt\omega}\), if \(\large \mu\hat{~}0\preceq\eta\) then \(\large \{\varphi(x,b_{\eta}),~\varphi(x,b_{\mu\hat{~}1})\}\) is inconsistent.

                    A theory is NTP1 (resp., NSOP1, NSOP2) if no formula has TP1 (resp., SOP1, SOP2).

                    It is clear that any NSOP1 theory must be NSOP2, and easy to show that NSOP2 is equivalent to NTP1 (this is implicit in ; the proof from is given here). The equivalence of NSOP1 and NSOP2 was open for many years, until proved in .

                    Remarks:
                    In analogy to forking in simple theories, there is a canonical notion of independence in NSOP1 theories called Kim-forking. NSOP1 can be characterized by symmetry or transitivity or local character of Kim-nonforking. See , , , .

                    SOP2 is shown to be maximal in Keisler's order in , and also characterizes maximality in the \(\large \triangleleft^*\)-order (, , ).

                    removed
                      NSOP3
                      NSOP3
                      no proper examples known
                        Fix a formula \(\large \varphi(x,y)\) with \(\large |x|=|y|\). Then \(\large \varphi(x,y)\) has the 3-strong order property (SOP3) if the set \[\large \{\varphi(x_1,x_2),\varphi(x_2,x_3),\varphi(x_3,x_1)\}\] is inconsistent, and there is a sequence \(\large (a_i)_{i\lt\omega}\) such that \(\large \varphi(a_i,a_j)\) holds for all \(\large i\lt j \).

                        A theory is NSOP3 if no formula has SOP3.

                        See , .
                        NSOP4
                        NSOP4
                          Fix a formula \(\large \varphi(x,y)\) with \(\large |x|=|y|\). Then \(\large \varphi(x,y)\) has the 4-strong order property (SOP4) if the set \[ \{\varphi(x_1,x_2),\varphi(x_2,x_3),\varphi(x_3,x_4),\varphi(x_4,x_1)\}\] is inconsistent, and there is a sequence \(\large (a_i)_{i\lt\omega}\) such that \(\large \varphi(a_i,a_j)\) holds for all \(\large i\lt j \).

                          A theory is NSOP4 if no formula has SOP4.

                          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
                          NSOPn+1
                            Fix n ≥ 3 and a formula \(\large \varphi(x,y)\) with \(\large |x|=|y|\). Then \(\large \varphi(x,y)\) has the n-strong order property (SOPn) if the set \[ \{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}\] is inconsistent, and there is a sequence \(\large (a_i)_{i\lt\omega}\) such that \(\large \varphi(a_i,a_j)\) holds for all \(\large i\lt j \).

                            A theory is NSOPn if no formula has SOPn.

                            See , .

                            The above definition of SOPn works equally well when n is 1 or 2, and in both cases is equivalent to well known properties (of the theory). When n is 1, the condition is equivalent to having an infinite model; and when n is 2, the condition is equivalent to having the order property (instability). However, the acronyms SOP1 and SOP2 are not defined this way in the literature, but rather are used for certain variations of the tree property. See the NTP1 (NSOP1/NSOP2) region for further details.
                            NFSOP
                            NFSOP
                              A type \(\large p(x,y)\), with \(\large |x|=|y|\), has the strong order property if there are \(\large (a_i)_{i\in\omega}\) such that \(\large p(a_i,a_j)\) holds for all \(\large i\lt j\), 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 \(\large n\ge 2\).

                              A theory is NFSOP (no finitary strong order property) if no type \(\large p(x,y)\) has the strong order property with \(\large |x|\) 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
                              NSOP
                                A formula \(\large \varphi(x,y)\) has the strict order property (SOP) if there are \(\large (a_i)_{i\lt\omega}\) 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 , .

                                Remarks:
                                The following characterizations of the strict order property are worth noting.
                                • T has SOP if and only if there is a formula \(\large \psi(y_1,y_2)\) (with \(\large |y_1|=|y_2|\)) defining a preorder with infinite chains. (In the above definition, take \(\large \psi(y_1,y_2)\) to be\(\large \forall x(\varphi(x,y_1)\rightarrow\varphi(x,y_2))\).)
                                • T has SOP if and only if there is a definable set X (in a saturated model) and an automorphism σ such that σ(X) properly contains X.
                                NTP2
                                NTP2
                                  A formula \(\large \varphi(x,y)\) has the tree property of the second kind (TP2) if there are \(\large (a_{i,j})_{i,j\lt\omega}\) such that
                                  • for all \(\large \sigma\in\omega^\omega\), \(\large \{\varphi(x,a_{n,\sigma(n)}):n\lt\omega\}\) is consistent.
                                  • for all \(\large n\lt\omega\) and \(\large i\lt j\lt\omega\), \(\large \{\varphi(x,a_{n,i}),~\varphi(x,a_{n,j})\}\) is inconsistent.
                                  A theory is NTP2 if no formula has the tree property 2.

                                  See , .

                                  distal
                                  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 it is NIP and all generically stable Keisler measures are smooth.
                                    • (Strong honest definitions) An NIP theory T is distal if and only if for any formula \(\large \varphi(x,y)\) there is a formula \(\large \psi(x,y_1,\ldots,y_n)\), with \(\large |y_i|=|y|\), such that for any finite set B of \(\large y\)-tuples, with |B|≥ 2, and any \(\large x\)-tuple \(\large a\), there are \(\large b_1,\ldots,b_n\) in B such that \(\large \psi(x,b_1,\ldots,b_n)\vdash\textrm{tp}_{\varphi}(a/B)\).
                                    removed
                                      strongly minimal
                                      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, any strongly minimal theory is ω-stable.
                                        Click a property above to highlight region and display details. Or click the map for specific region information.
                                        Reset
                                        List of Examples
                                          Implications Between Properties
                                          Click 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 contains no known examples, due to the following open questions:
                                          • Is the implication:
                                            SOP3 ⇒ TP1 (SOP1/SOP2)
                                            proper? See , , .
                                          • Is there a theory which is NSOP and NTP2, but not simple? See Exercise III.7.12 in , and further discussion on the fff blog.
                                          Open Examples
                                          The following examples are theories whose exact positions on the map are unknown. Click an example to see its definition, and to highlight the possible regions it could be in.
                                          • \(\large (\mathbb{C}(t),+,\cdot,0,1)\)
                                          • Artin braid groups
                                          NIP and NSOP
                                          NIP and SOP
                                          IP and NSOP
                                          IP and SOP
                                          infinite sets
                                          E.g. take the theory of \(\large \mathbb{N}\) in the empty language.

                                          characterization of forking
                                          This theory has quantifier elimination, so definable sets (in one variable) are finite or cofinite.

                                          See .
                                          ACF
                                          algebraically closed fields

                                          Axiomatized by the field axioms (in \(\large \mathcal{L}=\{+,\cdot,0,1\}\)), and the infinite axiom scheme saying every polynomial has a root.

                                          characterization of forking
                                          The theory can be completed by specifying the characteristic. By quantifier elimination, definable sets (in one variable) are Boolean combinations of polynomial zerosets, and so finite or cofinite.

                                          See .
                                          \(\large \mathbb{Q}\)-vector spaces
                                          Axiomatized by the axioms for torsion-free divisible abelian groups in \(\large \mathcal{L}=\{+,0\}\).

                                          characterization of forking
                                          By quantifier elimination, definable sets (in one variable) are given as Boolean combinations of formulas of the form nx = a, where n is an integer and a is an arbitrary element. Since the group is torsion free, such sets are finite or cofinite.

                                          See .
                                          DCF0
                                          differentially closed fields of characteristic 0

                                          characterization of forking
                                          Shown to be ω-stable by Blum.

                                          See .
                                          everywhere infinite forest
                                          Fraïssé limit of finite trees

                                          characterization of forking
                                          See , .
                                          \(\large (\mathbb{Z},+,0)\)
                                          characterization of forking
                                          See .
                                          finitely refining equivalence relations
                                          Infinitely many equivalence relations E0, E1,… such that E0 has two infinite classes, and each En-class is partitioned into two infinite En+1 classes.
                                          See , .
                                          infinitely refining equivalence relations
                                          Infinitely many equivalence relations E0, E1,… such that E0 has infinitely many infinite classes, and each En-class is partitioned into infinitely many infinite En+1 classes.

                                          characterization of forking
                                          See , .
                                          DCFp
                                          differentially closed fields of characteristic p

                                          characterization of forking
                                          Shown to be stable, but not superstable, by Shelah (1973).

                                          See .
                                          free group on n > 1 generators
                                          Shown to be stable by Sela (2006) . Non-superstability was proved much earlier by Gibone, and communicated by Poizat (see also ).
                                          RCF
                                          real closed fields

                                          Axiomatized by the ordered field axioms (in \( \mathcal{L}=\{+,\cdot,0,1,\lt\}\)), an axiom saying that every positive element has a square root, and an infinite axiom scheme saying every polynomial of odd degree has a root.
                                          See .
                                          \(\large (\mathbb{Z},\lt)\)
                                          Theory of discrete linear orders.
                                          See .
                                          \(\large (\mathbb{Q},\lt)\)
                                          Theory of dense linear orders without endpoints.
                                          See .
                                          \(\large (\mathbb{Z},+,\lt,0,1)\)
                                          Also known as Presburger Arithmetic.
                                          See .
                                          \(\large (\mathbb{Z},~x\mapsto x+1)\)
                                          characterization of forking
                                          By quantifier elimination, definable sets (in one variable) are of Boolean combinations of formulas of the form x = a + n, where a is an arbitrary element and n is an integer. Since successors are unique, such sets are finite or cofinite.

                                          See .
                                          ACVF
                                          algebraically closed valued fields

                                          See .
                                          \(\large (\mathbb{Q}_p,+,\cdot,v(x)\geq v(y))\)
                                          p-adic field with valuation
                                          See .
                                          random graph
                                          Axiomatized in the graph language by sentences saying that if A and B are finite disjoint sets of vertices then there is a vertex connected to everything in A and nothing in B.

                                          characterization of forking
                                          See , .
                                          pseudo-finite fields
                                          A field F is pseudo-finite if every sentence (in the field language) true of F holds in some finite field.
                                          See .
                                          ACFA
                                          algebraically closed fields with a generic automorphism

                                          characterization of forking
                                          See .
                                          \(\large \mathbb{Q}\)ACFA
                                          Model companion of the theory of fields with a \(\large (\mathbb{Q},+)\)-action.

                                          characterization of forking
                                          See .
                                          ultraproduct of \(\large \mathbb{Q}_p\)
                                          \(\large \mathbb{Q}_p\) denotes the field of p-adic numbers
                                          See .
                                          generic Kn-free graph
                                          Axiomatized in the graph language by sentences stating that if A and B are finite disjoint sets of vertices, with A Kn-1-free, then there is a vertex connected to everything in A and nothing in B.

                                          characterization of forking
                                          Given n ≥ 3, Kn denotes the complete graph on n vertices.

                                          See , .
                                          \(\large (\mathbb{Q},\textrm{cyc})\)
                                          The cyclic order on the rationals is the ternary relation cyc(a,b,c), which holds if and only if a<b<c or b<c<a or c<a<b.
                                          See .
                                          SCFp
                                          n
                                          separably closed fields of characteristic p and Eršov invariant n ≤ ∞
                                          See .
                                          \(\large (\mathbb{R},+,\cdot,\lt,0,1,\textrm{exp})\)
                                          the real exponential field
                                          Shown to be o-minimal by Wilkie (1996).

                                          See .
                                          \(\large ((\mathbb{Z}/4\mathbb{Z})^\omega,+)\)
                                          Totally categorical of Morley rank 2.

                                          See , .
                                          finitely cross-cutting equivalence relations
                                          Infinitely many equivalence relations E0, E1,… such that E0 has two infinite classes, and for all n < ω and I ⊆ {1,…,n}, the following axiom holds: \[\large \forall x\exists y\bigwedge_{i=1}^nE_i(x,y)^{i\in I}\]
                                          See .
                                          universal bowtie-free graph
                                          Complete theory of the universal and existentially closed countable graph omitting a 'bowtie' (sum of two triangles sharing a single vertex).
                                          See , .
                                          universal graph omitting odd cycles of length ≤ n
                                          Given odd n ≥ 3, the complete theory of the universal and existentially closed countable graph omitting all odd cycles of length at most n.
                                          See , .
                                          universal directed graph omitting cycles of length ≤ n
                                          Complete theory of the universal and existentially closed countable directed graph omitting directed cycles of length ≤ n.
                                          See .
                                          ZFC
                                          Theory of set theory in the language \(\large \mathcal{L}=\{\in\}\).
                                          \(\large (\mathbb{Z},+,\cdot,0,1)\)
                                          Complete theory of the ring of integers (a completion of Peano Arithmetic).
                                          \(\large (\mathbb{Q}^n,\lt_1,\ldots,\lt_n)\)
                                          Complete theory of \(\large \mathbb{Q}^n\), for n > 1, with coordinate orderings. In particular, given \(\large x,y\in\mathbb{Q}^n\), set \[\large x\lt_i y ~\Leftrightarrow~ x_i\lt y_i. \]
                                          See .
                                          Urysohn sphere
                                          The unique universal and ultrahomogeneous separable metric space (with distances bounded by 1).

                                          characterization of forking
                                          There are few options for what language to use for this structure. One is a relational language with binary relations \(\large d(x,y)\leq r\), for \(\large r\in \mathbb{Q}\cap[0,1]\). Alternatively, consider the Urysohn sphere as a metric structure in continuous logic with the empty language.

                                          See , .
                                          free nth root of the complete graph
                                          The unique universal and ω-homogeneous countable metric space with distances in {0,1,...,n}.

                                          characterization of forking
                                          This theory was originally defined with graph relations Ri for 0 ≤ i ≤ n, where Ri(x,y) holds if and only if x and y can be connected by a path of length i. The graph structure and metric structure can be identified via the equation

                                          d(x,y) = min{i : Ri(x,y)}.

                                          See , .
                                          Hrushovski's new strongly minimal set
                                          Hrushovski constructed an example of a strongly minimal theory, which is not locally modular and does not interpret an infinite group. This disproved Zilber's conjecture that a strongly minimal theory must either be locally modular or interpret an infinite field.
                                          See , .
                                          infinitely cross-cutting equivalence relations
                                          Infinitely many equivalence relations E0, E1,… such that each En has infinitely many infinite classes, and for all m < n < ω every En-class intersects every Em-class in an infinite set.
                                          See .
                                          non-simple generic limit of \(\large (\mathcal{K}_f,\leq)\) for good \(\large f\)
                                          \(\large (\mathcal{K}_f,\leq)\) is a class of finite structures (in a finite relational language), which is closed under free amalgamation and is equipped with a predimension and a control function \(\large f\).

                                          \(\large f\) is good if \(\large \mathcal{K}_f\) is closed under free \(\large \leq\)-amalgamation.

                                          We consider the theory of the generic structure \(\large \mathcal{M}_f\). These theories are always NSOP4, and the non-simple case is always SOP3 and TP2. The simple case can also be characterized by the closure of \(\large \mathcal{K}_f\) under independence theorem diagrams.
                                          See .
                                          VFA0
                                          The limit theory (as p→∞) of the Frobenius automorphism acting on an algebraically closed valued field of characteristic p.
                                          See .
                                          atomless Boolean algebras
                                          The theory of the Fraissé limit of Boolean algebras in the language \(\large \{0,1,\neg,\wedge,\vee\}\).
                                          See .
                                          generic Kr
                                          n                                          -free r-graph
                                          Given n > r > 2, Kr
                                          n                                           denotes the complete r-graph on n vertices.

                                          characterization of forking
                                          See .
                                          a strictly stable superflat graph
                                          Given integers m and n, let Km
                                          n                                           be the class of graphs obtained from the complete graph on n vertices by replacing each edge with a path containing at most m new vertices.

                                          A graph G is superflat if for all m there is some n such that G omits Km
                                          n                                          . G is ultraflat if there is some n such that for all m, G omits Km
                                          n                                          .

                                          Define the following strictly stable superflat graph. Begin with vertex set ω<ω∪ωω. For each σ in ωω and m < ω, add a path from σ to σ|m containing m new vertices.

                                          characterization of forking
                                          Any superflat graph is stable; any ultraflat graph (and so any planar graph) is superstable. The graph above interprets infinitely refining equivalence relations, and so is strictly stable.

                                          See , , .
                                          imperfect bounded PAC fields
                                          A variety over a field K is absolutely irreducible if it is not the union of two algebraic sets over an algebraically closed extension of K.

                                          A field K is pseudo-algebraically closed (PAC) if every absolutely irreducible variety defined over K has a K-rational point.

                                          K is perfect if either char(K)=0 or Kchar(K) = K. K is bounded if for all n > 1, K has finitely many separably algebraic extensions of degree n.

                                          characterization of forking
                                          If K is a PAC field then Th(K) is simple if and only if K is bounded. The theory of a perfect bounded PAC field is supersimple.

                                          See , , .
                                          Tfeq
                                          Consider the language \(\large \mathcal{L}=\{P,Q,E\}\), where \(\large P,Q\) are unary relations and \(\large E\) is a ternary relation. Tfeq is the model completion of the following theory:
                                          Shown to be NSOP1 in (this was first claimed in , but errors were found in the proof). See also , , .
                                          ω-free PAC fields
                                          A variety over a field K is absolutely irreducible if it is not the union of two algebraic sets over an algebraically closed extension of K.

                                          A field K is pseudo-algebraically closed (PAC) if every absolutely irreducible variety defined over K has a K-rational point.

                                          A PAC field K is ω-free if there is an elementary substructure L of K, whose absolute Galois group Aut(Lsep/L) is isomorphic to \(\large \hat{F}_\omega\), which is defined below.

                                          Let \(\large F_\omega\) be the free group on countably many generators. Let \(\large \mathcal{N}\) be the family of normal, finite-index subgroups of \(\large F_\omega\) containing cofinitely many generators. Then \[\large \hat{F}_{\!\omega}=\varprojlim_{N\in\mathcal{N}}F_\omega/N\]
                                          characterization of forking
                                          Shown to be NSOP1 in . See also , , .
                                          infinite-dimensional vector spaces with a bilinear form
                                          Defined in the 2-sorted language of vector spaces over an algebraically closed field.
                                          Shown to be NSOP1 in . See also , .
                                          densely ordered random graph
                                          Model completion of the theory of ordered graphs.
                                          See , .
                                          bounded pseudo real closed fields
                                          A field K, of characteristic 0, is bounded if, for every n, K has only finitely many extensions of degree n.

                                          A variety over a field K is absolutely irreducible if it is not the union of two algebraic sets over an algebraically closed extension of K.

                                          A field K is pseudo real closed if, for every absolutely irreducible variety V defined over K, if V has a Kr-rational point for every real closure Kr of K, then V has a K-rational point.
                                          We assume here that the pseudo real closed field K is not real closed or algebraically closed, which ensures the independence property, and also that K has at least one order, which ensures the strict order property (a pseudo real closed field with no orders is pseudo algebraically closed).

                                          See .
                                          \(\large (\mathbb{Z}^{\omega},+,0)\)
                                          For a fixed prime p, the subgroups Hn of elements divisible by pn, for n ≥ 0, form a descending chain of definable subgroups, each of which is infinite index in the previous. Therefore the theory is strictly stable.
                                          See , .
                                          \(\large (\mathbb{Z},+,0,\Gamma)\)
                                          Complete theory of the abelian group \(\large (\mathbb{Z},+,0)\) expanded by a predicate for a fixed finitely generated multiplicative submonoid \(\large \Gamma\) of \(\large \mathbb{Z}^+\).
                                          Superstable of U-rank ω, and not dp-minimal.

                                          See or for the case of one generator, and for the general case.

                                          The failure of dp-minimality can be established using an elementary direct argument similar to Proposition 2.7 of . More generally, by , there are no proper stable expansions of \(\large (\mathbb{Z},+,0)\) with finite dp-rank.
                                          extra-special p-group
                                          Given an odd prime p, there is a group G satisfying the following properties: Such a group G is called an extra-special p-group. The defining properties are expressible in the language of groups, and determine a complete theory.
                                          Supersimple of SU-rank 1, and unstable.

                                          See .
                                          \(\large (\mathbb{T},+,\cdot,0,1,\partial,\leq,\preccurlyeq)\)
                                          ordered valued differential field of logarithmic-exponential transseries

                                          \(\large \mathbb{T}\) is a model of the theory Tnl of ω-free newtonian Liouville closed H-fields, which is the model companion of the theory of H-fields. The theory Tnl has two completions: one with small derivation of which \(\large \mathbb{T}\) is a model, and one which does not have small derivation.
                                          Both completions of Tnl are NIP, unstable, and not dp-minimal.

                                          See .
                                          \(\large (\mathbb{Z},+,\le_p,0,1)\)
                                          Given a prime p, let ≤p denote the pre-order on integers induced from the p-valuation: x≤y if and only if vp(x)≤vp(y).
                                          See .
                                          \(\large (\mathbb{Z},+,0,\text{Sqf})\)
                                          Complete theory of the abelian group \(\large (\mathbb{Z},+,0)\) expanded by a predicate for the set Sqf of squarefree integers.
                                          Supersimple of SU-rank 1 and unstable.

                                          See . This result builds on which, assuming Dickson's Conjecture, gives the same classification of the expansion of \(\large (\mathbb{Z},+,0)\) by a predicate for the primes and their negatives.
                                          multicolored directed graphs omitting directed cycles
                                          Consider a language with countably many binary relation symbols (Rn). Let T be the model completion of the theory expressing that each Rn is a directed graph relation with no directed n-cycles.
                                          TP2, NSOP and SOP∞.

                                          See .
                                          \(\large (\mathbb{N},\cdot)\)
                                          Also known as Skolem Arithmetic.
                                          Skolem arithmetic is decidable.

                                          See .
                                          generic binary function
                                          The model completion of the empty theory in a language containing only one binary function symbol.

                                          characterization of forking
                                          Given a first-order language \(\large \mathcal{L}\), the empty \(\large \mathcal{L}\)-theory has a model completion, T0, whose completions are determined by specifying the diagram of the constant symbols in \(\large \mathcal{L}\). Suppose T is a completion of T0. Then: See .
                                          generic Km,n-free bipartite graph
                                          The model completion of the theory of Km,n-free bipartite graphs, for m,n ≥ 2.

                                          characterization of forking
                                          This theory is considered in the language of bipartite graphs consisting of a binary graph relation I, together with unary predicates P and L for the bipartition. Alternatively, one can view P and L as sorts for abstract points and lines, and I as an incidence relation. For example, if m = n = 2, then this is the model completion of the theory of combinatorial projective planes.

                                          See .
                                          Henson digraphs
                                          Given a set F of finite tournaments, the the class of F-free directed graphs is a Fraïssé class with ℵ0-categorical Fraïssé limit.
                                          Constructed by Henson to exhibit continuum many pairwise nonisomorphic homogeneous directed graphs.

                                          See .
                                          \(\large (\mathbb{R},+,\cdot,2^{\mathbb{Q}})\)
                                          The real field expanded by a predicate for \(\large 2^{\mathbb{Q}}\).
                                          One can replace \(\large 2^{\mathbb{Q}}\) by any dense subgroup of \(\large \mathbb{R}^+\) with the Mann property, or by a dense transcendence basis for \(\large \mathbb{R}\).

                                          See .
                                          generic Steiner triple system
                                          The Fraïssé limit of Steiner triple systems in the language of quasigroups.
                                          A Steiner triple system is a set X together with a collection B of three-element subsets of X (called "blocks"), with the property that any two elements of X lie in a unique block in B. If X is a Steiner triple system, then there is a quasigroup operation on X, which sends pairs (x,x) to x, and sends pairs (x,y), with x and y distinct, to the third point on the block determined by x and y.

                                          See .
                                          a non-definably-amenable group in a simple theory
                                          Let K be an algebraically closed field of characteristic zero. Then there is a partition

                                          SL2(K) = C1 ∪ C2 ∪ C3 ∪ C4

                                          such that the theory of

                                          G := (SL2(K),⋅,C1,C2,C3,C4)

                                          is supersimple, and for all 1≤i≤4, SL2(K) can be covered by three left translates of Ci.

                                          The conditions on the Ci's imply that there is no left-invariant finitely additive probability measure on the definable subsets of G, i.e., G is not definably amenable.
                                          See .
                                          ACFpG
                                          Generic expansion of the theory of algebraically closed fields of characteristic p > 0 by an additive subgroup.
                                          This theory is the model companion of ACFp expanded by a predicate for an additive subgroup.

                                          See .
                                          \(\large (\mathbb{C}(t),+,\cdot,0,1)\)
                                          The field of rational functions over the complex numbers.
                                          This field is not superstable, since any infinite superstable field is algebraically closed ; and it is not dp-minimal (see for a classification of dp-minimal fields). The stable fields conjecture asserts that every infinite stable field is separably closed. \(\large \mathbb{C}(t)\) has become a canonical test case for this conjecture.

                                          list of open examples
                                          Farey graph
                                          The vertex set of the Farey graph is the projective rational line (i.e., all rational numbers, and a point at infinity identified as 1/0). Two vertices a/b and c/d (in lowest terms) are connected by an edge if and only if |ad - bc| = 1.

                                          picture of the Farey graph

                                          characterization of forking
                                          This graph is planar, and thus ultraflat, which implies it is superstable and dp-minimal. It is not strongly minimal since every vertex has an infinite and co-infinite neighborhood. For ω-stability, see .

                                          See , , .
                                          Artin braid groups
                                          Given n > 2, the braid group Bn is the group with the following presentation

                                          \( \langle x_1,\ldots,x_{n-1}\mid x_ix_{i+1}x_i=x_{i+1}x_ix_{i+1}\rangle\)
                                          The center of Bn is an infinite cylic group, and thus Bn is not ω-stable. Bn contains the free group on 2 generators as a subgroup, and thus is not dp-minimal. It also unknown whether Bm and Bn are elementarily equivalent when m ≠ n.

                                          list of open examples
                                          a strictly stable expansion of \(\large (\mathbb{Z},+,0)\)
                                          Let \(\large Q\) be the set of factorials and let \(\large \mathcal{Q}\) be an arbitrary structure with universe \(\large Q\). Consider the expansion \(\large (\mathbb{Z},+,0,\mathcal{Q})\) of \(\large (\mathbb{Z},+,0)\) obtained by naming \(\large Q\) with a unary predicate and enriching it with the structure \(\large \mathcal{Q}\). Then \(\large (\mathbb{Z},+,0,\mathcal{Q})\) is strictly stable if and only if \(\large \mathcal{Q}\) is. (The same holds for superstable, NIP, NTP2, simple, and NSOP1.)

                                          See .
                                          One can accomplish this with an expansion by a single unary predicate as follows. Suppose \(\large \mathcal{Q}\) is a graph \(\large (Q,E)\). Define \[\large A=Q\cup \{x+y:(x,y)\in E\}. \] Then \(\large (\mathbb{Z},+,0,A)\) is interdefinable with \(\large (\mathbb{Z},+,\mathcal{Q})\). For a strictly stable example, let \(\large \mathcal{Q}\) be a countable model of the graph defined here. Or one can construct a graph bi-interpretable with any given structure in a finite language (see Theorem 5.5.1 in ).

                                          The failure of dp-minimality (which happens even in the reduct \(\large (\mathbb{Z},+,0,Q)\)) can be established using an elementary direct argument similar to Proposition 2.7 of . More generally, by , there are no proper stable expansions of \(\large (\mathbb{Z},+,0)\) with finite dp-rank.
                                          strongly minimal
                                          o-minimal
                                          ω-stable, dp-minimal, and not strongly minimal
                                          superstable, dp-minimal, and not ω-stable
                                          stable, dp-minimal, and not superstable
                                          dp-minimal, distal, and not o-minimal
                                          NIP, SOP, dp-minimal, and not distal
                                          NIP and SOP, but not dp-minimal or distal
                                          IP, SOP, NTP2
                                          supersimple and unstable
                                          simple and unstable, but not supersimple
                                          NTP2, TP1 (SOP1/SOP2), and NSOP3
                                          unknown

                                          It is unknown whether the implication SOP3 ⇒ TP1 is strict.
                                          NTP2, SOP3, and NSOP4
                                          unknown

                                          It is unknown whether the strong order property hierarchy is strict inside NTP2.
                                          NTP2, SOPn, and NSOPn+1 for some n ≥ 4
                                          unknown

                                          It is unknown whether the strong order property hierarchy is strict inside NTP2.
                                          NTP2, NFSOP, and SOPn for all
                                          n ≥ 3
                                          unknown

                                          It is unknown whether the strong order property hierarchy is strict inside NTP2.
                                          NTP2, NSOP, and FSOP
                                          unknown

                                          It is unknown whether the strong order property hierarchy is strict inside NTP2.
                                          NTP1 (NSOP1/NSOP2) and TP2
                                          TP2, TP1 (SOP1/SOP2), and NSOP3
                                          unknown

                                          It is unknown whether the implication SOP3 ⇒ TP1 is strict.
                                          TP2, SOP3, and NSOP4
                                          TP2, SOPn, and NSOPn+1 for some n ≥ 4
                                          To keep this section distinct from the others, we assume n ≥ 4. However every known example of a theory that is SOPn and NSOPn+1 for some n ≥ 4, has some analogue for any n ≥ 3.
                                          TP2, NFSOP, and SOPn for all
                                          n ≥ 3
                                          TP2, FSOP, and NSOP
                                          TP2 and SOP
                                          distal and not dp-minimal
                                          ω-stable, and not dp-minimal
                                          superstable, not ω-stable, and not dp-minimal
                                          stable, not superstable, and not dp-minimal