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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 AM.

    See , , .
    superstable
    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
      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
        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
          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
            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
              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
                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
                  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
                    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
                      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
                        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
                          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
                            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
                              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
                                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
                                  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
                                      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.
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                                        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.
                                          NIP and NSOP
                                          NIP and SOP
                                          IP and NSOP
                                          IP and SOP
                                          infinite sets
                                          E.g. take the theory of \(\mathbb{N}\) 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 \(\mathcal{L}=\{+,\cdot,0,1\}\)), 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 .
                                          \(\mathbb{Q}\)-vector spaces
                                          Axiomatized by the axioms for torsion-free divisible abelian groups in \(\mathcal{L}=\{+,0\}\).

                                          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 , .
                                          \((\mathbb{Z},+,0,1)\)
                                          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, but not superstable, by Sela (2006).

                                          See .
                                          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 .
                                          \((\mathbb{Z},\lt)\)
                                          Theory of discrete linear orders.
                                          See .
                                          \((\mathbb{Q},\lt)\)
                                          Theory of dense linear orders.
                                          See .
                                          \((\mathbb{Z},+,\lt,0,1)\)
                                          Also known as Presburger Arithmetic.
                                          See .
                                          \((\mathbb{Z},~x\mapsto x+1)\)
                                          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 .
                                          \((\mathbb{Q}_p,+,\cdot,v(x)\geq v(y))\)
                                          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 .
                                          \(\mathbb{Q}\)ACFA
                                          Model companion of the theory of fields with a \((\mathbb{Q},+)\)-action.

                                          characterization of forking
                                          See .
                                          ultraproduct of \(\mathbb{Q}_p\)
                                          \(\mathbb{Q}_p\) 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 , .
                                          \((\mathbb{Q},\textrm{cyc})\)
                                          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 .
                                          \((\mathbb{R},+,\cdot,\lt,0,1,\textrm{exp})\)
                                          the real exponential field
                                          Shown to be o-minimal by Wilkie (1996).

                                          See .
                                          \(((\mathbb{Z}/4\mathbb{Z})^\omega,+)\)
                                          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: \[\forall x\exists y\left(\bigwedge_{i\in I}E_i(x,y)\wedge\bigwedge_{i\in n\backslash I}\neg E_i(x,y)\right)\]
                                          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 \(\mathcal{L}=\{\in\}\).
                                          \((\mathbb{Z},+,\cdot,0,1)\)
                                          Complete theory of the ring of integers (a completion of Peano Arithmetic).
                                          \((\mathbb{Q}^n,\lt_1,\ldots,\lt_n)\)
                                          Complete theory of \(\mathbb{Q}^n\), for n > 1, with coordinate orderings. In particular, given \(x,y\in\mathbb{Q}^n\), set \[x\lt_i y ~\Leftrightarrow~ x_i\lt y_i. \]
                                          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 to use distance relations \(d(x,y)\leq r\), for \(r\in\mathbb{Q}\). Alternatively, one consider the Urysohn sphere as a metric structure in continuous logic.

                                          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 ≤ in, 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 thought of as the same via
                                          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 n < ω each En+1 class splits each En class into infinitely many pieces.
                                          See .
                                          non-simple generic limit of \((\mathcal{K}_f,\leq)\) for good \(f\)
                                          \((\mathcal{K}_f,\leq)\) 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 \(f\).

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

                                          We consider the theory of the generic structure \(\mathcal{M}_f\). 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 \(\mathcal{K}_f\) 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 \(\{0,1,\neg,\wedge,\vee\}\).
                                          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∪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 defined over a field K is absolutely irreducible if it is not the union of two algebraic sets defined over some 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 it has characteristic 0, or if K has characteristic p then Kp = 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 \(\mathcal{L}=\{P,Q,E\}\), where \(P,Q\) are unary relations and \(E\) is a ternary relation. Tfeq is the model completion of the following theory:
                                          • \(P\) and \(Q\) form a partition.
                                          • For all \(a\) in \(P\), \(E(a,x,y)\) is an equivalence relation on \(Q\).
                                          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 defined over a field K is absolutely irreducible if it is not the union of two algebraic sets defined over some 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 \(\hat{F}_\omega\), which is defined below.

                                          Let \(F_\omega\) be the free group on countably many generators. Let \(\mathcal{N}\) be the family of normal, finite-index subgroups of \(F_\omega\) containing cofinitely many generators. Then \[\hat{F}_\omega=\varprojlim_{N\in\mathcal{N}}F_\omega/N\]
                                          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 defined over a field K is absolutely irreducible if it is not the union of two algebraic sets defined over some 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 .
                                          \((\mathbb{Z}^{\omega},+,0)\)
                                          Complete theory of the abelian group \((\mathbb{Z}^{\omega},+,0)\).
                                          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 , .
                                          \((\mathbb{Z},+,0,\Gamma)\)
                                          Complete theory of the abelian group \((\mathbb{Z},+,0)\) expanded by a predicate for a fixed finitely generated multiplicative submonoid \(\Gamma\) of \(\mathbb{Z}^+\).
                                          Superstable of U-rank ω, and not dp-minimal.

                                          See , for the case of one generator, and for the general case.
                                          extra-special p-group
                                          Given an odd prime p, there is a group G satisfying the following properties:
                                          • G is infinite,
                                          • every nontrivial element of G has order p,
                                          • Z(G) is cyclic of order p and equals G'.
                                          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 .
                                          \((\mathbb{T},+,\cdot,0,1,\partial,\leq,\preccurlyeq)\)
                                          ordered valued differential field of logarithmic-exponential transseries

                                          \(\mathbb{T}\) 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 \(\mathbb{T}\) is a model, and one which does not have small derivation.
                                          Both completions of Tnl are NIP, unstable, and not dp-minimal.

                                          See .
                                          \((\mathbb{Z},+,\le_p,0,1)\)
                                          Given a prime p, let ≤p denote the pre-order on integers induced from the p-valuation: xy if and only if vp(x)≤vp(y).
                                          • Dp-minimal, unstable, and not o-minimal.
                                          • Expanding by the preorders associated to n distinct primes yields dp-rank n.
                                          See .
                                          \((\mathbb{Z},+,0,\text{Sqf})\)
                                          Complete theory of the abelian group \((\mathbb{Z},+,0)\) 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 \((\mathbb{Z},+,0)\) 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 .
                                          \((\mathbb{N},\cdot)\)
                                          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 \(\mathcal{L}\), the empty \(\mathcal{L}\)-theory has a model completion, T0, whose completions are determined by specifying the diagram of the constant symbols in \(\mathcal{L}\). Suppose T is a completion of T0. Then:
                                          • T is NSOP1.
                                          • If \(\mathcal{L}\) contains a function symbol of arity at least 2, then T is not simple.
                                          • If \(\mathcal{L}\) contains a relation symbol of arity at least 2, then T is unstable.
                                          • If all functions and relations in \(\mathcal{L}\) have arity at most 1, then T is superstable of U-rank 1.
                                          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 .
                                          \((\mathbb{R},+,\cdot,2^{\mathbb{Q}})\)
                                          The real field expanded by a predicate for \(2^{\mathbb{Q}}\).
                                          More generally, \(2^{\mathbb{Q}}\) can be replaced by any dense multiplicative subgroup of the positive reals with the Mann property, or by a dense transcendence basis for \(\mathbb{R}\) over \(\mathbb{Q}\).

                                          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 .
                                          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, and NSOP3
                                                                  unknown

                                                                  It is unknown whether or not the implication SOP3 ⇒ TP1 is strict.
                                                                  NTP2, SOP3, and NSOP4
                                                                    unknown

                                                                    It is unknown whether or not the strong order property hierarchy is strict inside NTP2.
                                                                    NTP2, SOPn, and NSOPn+1 for some
                                                                    n ≥ 4,
                                                                      unknown

                                                                      It is unknown whether or not the strong order property hierarchy is strict inside NTP2.
                                                                      NTP2, NFSOP, and SOPn for all n ≥ 1
                                                                        unknown

                                                                        It is unknown whether or not the strong order property hierarchy is strict inside NTP2.
                                                                        NTP2, NSOP, and FSOP
                                                                          unknown

                                                                          It is unknown whether or not the strong order property hierarchy is strict inside NTP2.
                                                                          NSOP1 and TP2
                                                                            NTP1 and SOP1
                                                                              unknown

                                                                              It is unknown whether or not the implication TP1 ⇒ SOP1 is strict.
                                                                              TP2, TP1, and NSOP3
                                                                                unknown

                                                                                It is unknown whether or not 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 ≥ 1
                                                                                      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