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.
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)
BACKGROUND In the Swedish society, as in many other societies, many children and adolescents with mental health problems do not receive the help they need. As the Swedish society becomes increasingly multicultural, and as ethnic and economic residential segregation become more pronounced, this study utilises ethnicity and neighbourhood context to examine… (More)