We consider a set K = n∈N Kn of nite structures such that all members of Kn have the same universe, the cardinality of which approaches ∞ as n → ∞. Each structure in K may have a nontrivial underlying pregeometry and on each Kn we consider a probability measure, either the uniform measure, or what we call the dimension conditional measure. The main… (More)
We study a class C of ℵ 0-categorical simple structures such that every M in C has an uncomplicated forking behavior and where definable relations in M which do not cause forking are independent in a sense that is made precise. The SU-rank of such M may be n for any natural number n > 0. The most well-known unstable structure in this class is the random… (More)
For any xed integer R ≥ 2 we characterise the typical structure of undi-rected graphs with vertices 1,. .. , n and maximum degree R, as n tends to innity. The information is used to prove that such graphs satisfy a labelled limit law for rst-order logic. If R ≥ 5 then also an unlabelled limit law holds.
A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of… (More)
For integers l ≥ 1, d ≥ 0 we study (undirected) graphs with vertices 1,. .. , n such that the vertices can be partitioned into l parts such that every vertex has at most d neighbours in its own part. The set of all such graphs is denoted Pn(l, d). We prove a labelled rst-order limit law, i.e., for every rst-order sentence ϕ, the proportion of graphs in… (More)
We work with a nite relational vocabulary with at least one relation symbol with arity at least 2. Fix any integer m > 1. For almost all nite structures (labelled or unlabelled) such that at least m elements are moved by some automor-phisms, the automorphism group is (Z2) i for some i ≤ (m + 1)/2; and if some relation symbol has arity at least 3, then the… (More)
Suppose that M is countable, binary, primitive, homogeneous, simple and 1-based. We prove that the SU-rank of the complete theory of M is 1. It follows that M is a random structure. The conclusion that M is a random structure does not hold if the binarity condition is removed, as witnessed by the generic tetrahedron-free 3-hypergraph. However, to show that… (More)
Suppose that M is an innite structure with nite relational vocabulary such that every relation symbol has arity at most 2. If M is simple and homogeneous then its complete theory is supersimple with nite SU-rank which cannot exceed the number of complete 2-types over the empty set.