Related papers: Independence over arbitrary sets in NSOP$_1$ theor…

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm{ACFG}$. This theory was introduced recently as a new example of $\mathrm{NSOP}_1$ non…

In this work we study some examples of groups definable and type-definable in NSOP1 theories. We exhibit some behaviors of these groups that differ from the ones of simple groups. We take interest in the notions of generics and stabilizers,…

We define weak stable Kim-forking, a notion that generalizes stable forking to the context of NSOP1 theories. We adapt some of the known results on stable forking to this context.

We presents an independence relation on sets, one can define dimension by it, assuming that we have an abstract elementary class with a forking notion that satisfies the axioms of a good frame minus stability.

Tree properties are introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP$_1$ or TP$_2$. In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion…

We try to understand complete types over a somewhat saturated model of a complete first order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory…

We give an example of an SOP theory $T$, such that any $L(M)$-formula $\varphi(x,y)$ with $|y|=1$ is NSOP. We show that any such $T$ must have the independence property. We also give a simplified proof of Lachlan's theorem that if every…

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if $p = tp(b/A)$ does not fork…

We study the theory $T_{m,n}$ of existentially closed incidence structures omitting the complete incidence structure $K_{m,n}$, which can also be viewed as existentially closed $K_{m,n}$-free bipartite graphs. In the case $m = n = 2$, this…

Monadic stability and the more general monadic dependence (or NIP) are tameness conditions for classes of logical structures, studied in the 80's in Shelah's classification program in model theory. They recently emerged in algorithmic and…

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…

Structural independence is the (conditional) independence that arises from the structure rather than the precise numerical values of a distribution. We develop this concept and relate it to $d$-separation and structural causal models.…

We introduce a family of local ranks DQ depending on a finite set Q of pairs of the form (\varphi(x,y),q(y)) where \varphi(x,y) is a formula and q(y) is a global type. We prove that in any NSOP1 theory these ranks satisfy some desirable…

We characterize nonforking (Morley) sequences in dependent theories in terms of a generalization of Poizat's special sequences and show that average types of Morley sequences are stationary over their domains. We characterize generically…

This course introduces the fruitful links between model theory and a combinatoric of sets given by independence relations. An independence relation on a set is a ternary relation between subsets. Chapter 1 should be considered as an…

Let T be a countable, small simple theory. In this paper, we prove for such T, the notion of Lascar Strong type coincides with the notion of a strong type,over an arbitrary set.

We give definitions of the properties OP, IP, $k$-TP, TP$_1$, $k$-TP$_2$, SOP$_1$, SOP$_2$ and SOP$_3$ in positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in…

We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences…

We give a category-theoretic construction of simple and NSOP$_1$-like independence relations in locally finitely presentable categories, and in the more general locally finitely multipresentable categories. We do so by identifying…

We develop a new notion of independence suggested by Scanlon (th-independence). We prove that in a large class of theories (which includes all simple theories) this notion has many of the properties needed for an adequate geometric…