Abstract. We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded forking assuming NTP 2 .

We study model theoretic tree properties (TP, TP 1 , TP 2 ) and their associated cardinal invariants (κ cdt , κsct, κ inp , respectively). In particular, we obtain a quantitative refinement of Shelah's theorem (TP ⇒ TP 1 ∨ TP 2 ) for countable theories, show that TP 1 is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak k − TP 1 is equivalent to TP 1 (answering a question of Kim and Kim). Besides, we give a characterization of NSOP 1 via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed NSOP 1 . arXiv:1505.00454v2 [math.LO]

Abstract. We initiate a systematic study of the class of theories without the tree property of the second kind -NTP 2 . Most importantly, we show: the burden is "sub-multiplicative" in arbitrary theories (in particular, if a theory has TP 2 then there is a formula with a single variable witnessing this); NTP 2 is equivalent to the generalized Kim's lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters -so the dp-rank of a 1-type in any theory is always witnessed by sequences of singletons; in NTP 2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic (0, 0) is NTP 2 (strong, of finite burden) if and only if the residue field is NTP 2 (the residue field and the value group are strong, of finite burden respectively), so in particular any ultraproduct of p-adics is NTP 2 ; adding a generic predicate to a geometric NTP 2 theory preserves NTP 2 .

We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.

We prove that externally definable sets in first order N IP theories have honest definitions, giving a new proof of Shelah's expansion theorem. Also we discuss a weak notion of stable embeddedness true in this context. Those results are then used to prove a general theorem on dependent pairs, which in particular answers a question of Baldwin and Benedikt on naming an indiscernible sequence.

It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into very homogeneous semialgebraic pieces up to a small error (e.g., see [33,2,16,18]). We show that similar results can be obtained for families of graphs with the edge relation uniformly definable in a structure satisfying a certain model theoretic property called distality, with respect to a large class of generically stable measures. Moreover, distality characterizes these strong regularity properties. This applies in particular to graphs definable in arbitrary o-minimal structures and in p-adics.1 2 ARTEM CHERNIKOV AND SERGEI STARCHENKO Theorem 1.2 (Erdős, Hajnal and Pach [14]). For any finite graph H there is a constant δ = δ(H) > 0 such that every H-free graph on n vertices has a homogeneous pairThe following definition is taken from [17].Remark 1.4. Is is shown in [2] that if a family of finite graphs G has the strong Erdős-Hajnal property and is closed under taking induced subgraphs then it has the Erdős-Hajnal property.In this paper we consider families of graphs whose edge relations are given by a fixed definable relation in a first-order structure.Definition 1.5. Let M be a first-order structure and R ⊆ M k × M k be a definable relation. Consider the family G R of all finite graphs V = (G, E) where G ⊆ M k is a finite subset and E = (V ×V )∩R. We say that R satisfies the (strong) Erdős-Hajnal property if the family G R does.We extend this notion to the bi-partite case. Definition 1.6. Let M be a first-order structure and R ⊆ M m × M n a definable relation. (1) A pair of subsets A ⊆ M m , B ⊆ M n is called R-homogeneous if either A × B ⊆ R or (A × B) ∩ R = ∅. (2) We say that the relation R satisfies the strong Erdős-Hajnal property if there is a constant δ = δ(R) > 0 such that for any finite subsets A ⊆ M m , B ⊆ M n there are A 0 ⊆ A, B 0 ⊆ B with |A 0 | ≥ δ|A|, |B 0 | ≥ δ|B|, and the pair A 0 , B 0 is R-homogeneous.Our motivation for this work comes from the following remarkable theorem by Alon et al.Theorem 1.7 ([2, Theorem 1.1]). If R ⊆ R n × R m is a semialgebraic relation then R has the strong Erdős-Hajnal property.Remark 1.8.(i) Although it is not stated explicitly in [2], but can be easily derived from the proof, homogeneous pairs in the above theorem can be chosen to be relatively uniformly definable. (ii) The above theorem was generalized by Basu (see [5]) to (topologically closed) relations definable in arbitrary o-minimal expansions of real closed fields.Besides the Erdős-Hajnal property for semialgebraic graphs, the above theorem has many other applications including unit distance problems [32], improved bounds in higher dimensional semialgebraic Ramsey theorem [11], [2, Theorem 1.2], algorithmic property testing [18], and can also be used to obtain a strong Szemerédi-type regularity lemma for semialgebraic graphs [16,18] (see also Section 5).The aim of this article is to demonstrate that the above r...

We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; over a model, a family of non-forking instances of a formula (with parameters ranging over a type-definable set) can be covered with finitely many invariant types; we give some criteria for the boundedness of an expansion by a new predicate in a distal theory; naming an arbitrary small indiscernible sequence preserves NIP, while naming a large one doesn't; there are models of NIP theories over which all 1-types are definable, but not all n-types.

In this note we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence (for 1 ≤ n < ω) recently introduced by Shelah [She07]. In the same way as dependence of a theory means its inability to encode a bipartite random graph with a definable edge relation, n-dependence corresponds to the inability to encode a random (n + 1)-partite (n + 1)-hypergraph with a definable edge relation. Most importantly, we characterize n-dependence by counting ϕ-types over finite sets (generalizing Sauer-Shelah lemma and answering a question of Shelah from [She05]) and in terms of the collapse of random ordered (n + 1)-hypergraph indiscernibles down to order-indiscernibles (which implies that the failure of n-dependence is always witnessed by a formula in a single free variable).