Let S∞ denote the topological group of permutations of the natural numbers. A closed subgroup G of S∞ is called oligomorphic if for each n, its natural action on n-tuples of natural numbers has onl...

We investigate possible cardinalities of maximal antichains in the poset of copies \,\subseteq \rangle \) of a countable ultrahomogeneous relational structure \. It turns out that if the age of \ has the strong amalgamation property, then, defining a copy of \ to be large iff it has infinite intersection with each orbit of \, the structure \ can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (...) antichains of size continuum in the poset \\). Finally, we show that the posets of copies of all countable ultrahomogeneous partial orders contain maximal antichains of cardinality continuum and determine which of them contain countable maximal antichains. That holds, in particular, for the generic poset. (shrink)

Let V be a finite relational vocabulary in which no symbol has arity greater than 2. Let be countable V‐structure which is homogeneous, simple and 1‐based. The first main result says that if is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the first, implies (without the assumption on primitivity) that if is “coordinatized” by a set with SU‐rank 1 and there is no definable (without parameters) nontrivial equivalence relation on (...) M with only finite classes, then is strongly interpretable in a random structure. (shrink)

The notion of a strongly determined type over A extending p is introduced, where p .S. A strongly determined extension of p over A assigns, for any model M )- A, a type q S extending p such that, if realises q, then any elementary partial map M → M which fixes acleq pointwise is elementary over . This gives a crude notion of independence which arises very frequently. Examples are provided of many different kinds of theories with strongly determined (...) types, and some without. We investigate a notion of multiplicity for strongly determined types with applications to ‘involved’ finite simple groups, and an analogue of the Finite Equivalence Relation Theorem. Lifting of strongly determined types to covers of a structure is discussed, and an application to finite covers is given. (shrink)

From classical, Fraïissé-homogeneous, ($\leq \omega$)-categorical theories over finite relational languages, we construct intuitionistic theories that are complete, prove negations of classical tautologies, and admit quantifier elimination. We also determine the intuitionistic universal fragments of these theories.

Hrushovski originated the study of “flat” stable structures in constructing a new strongly minimal set and a stable 0-categorical pseudoplane. We exhibit a set of axioms which for collections of finite structure with dimension function δ give rise to stable generic models. In addition to the Hrushovski examples, this formalization includes Baldwin's almost strongly minimal non-Desarguesian projective plane and several others. We develop the new case where finite sets may have infinite closures with respect to the dimension function δ. In (...) particular, the generic structure need not be ω-saturated and so the argument for stability is significantly more complicated. We further show that these structures are “flat” and do not interpret a group. (shrink)

We study finite ℓ‐colourable structures with an underlying pregeometry. The probability measure that is used corresponds to a process of generating such structures by which colours are first randomly assigned to all 1‐dimensional subspaces and then relationships are assigned in such a way that the colouring conditions are satisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure, where we now forget the specific colouring of the generating process, has (...) a given property. With this measure we get the following results: (1) A zero‐one law. (2) The set of sentences with asymptotic probability 1 has an explicit axiomatisation which is presented. (3) There is a formula (not directly speaking about colours) such that, with asymptotic probability 1, the relation “there is an ℓ‐colouring which assigns the same colour to x and y” is defined by. (4) With asymptotic probability 1, an ℓ‐colourable structure has a unique ℓ‐colouring (up to permutation of the colours). (shrink)