We introduce the notion of the burden of a partial type in a complete first-order theory and call a theory strong if all types have almost finite burden. In a simple theory it is the supremum of the weights of all extensions of the type, and a simple theory is strong if and only if all types have finite weight. A theory without the independence property is… (More)
A ternary relation | between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence… (More)
We present an updated exposition of the classical theory of complete first order theories without the independence property.
The notion of a VC-minimal theory is introduced, a slightly more general variant of C-minimality that also includes all strongly minimal or (weakly) o-minimal theories. The 1-dimensional definable sets in a VC-minimal theory have a good 'swiss cheese' representation similar to the C-minimal case. VC-minimal theories are dp-minimal; in particular they do not… (More)
We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thorn-forking, thorn-dividing, splitting or finite satisfiability share in all complete theories. We examine the relation between four additional… (More)
We give a simple proof that the straightforward generalisation of clique-width to arbitrary structures can be unbounded on structures of bounded tree-width. This can be corrected by allowing fusion of elements.
We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from  on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of p (m) for all m. We use the latter result to answer in full generality a question posed by Hasson and… (More)
A class of graphs is nowhere dense if for every integer r there is a finite upper bound on the size of cliques that occur as (topological) r-minors. We observe that this tameness notion from algorithmic graph theory is essentially the earlier stability theoretic notion of superflatness. For subgraph-closed classes of graphs we prove equivalence to stability… (More)
A theory with the strict order property does not eliminate hyperimaginaries. Hence a theory without the independence property eliminates hyperimaginaries if and only if it is stable. A type definable equivalence relation is an equivalence relation on tuples of a certain length (possibly infinite) which is defined by a partial type E(¯ x; ¯ y) over ∅. A… (More)