We give a general criterion for the (bounded) simplicity of the automorphism groups of certain countable structures and apply it to show that the isometry group of the Urysohn space modulo the normal subgroup of bounded isometries is a simple group.

This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions. The authors introduce the classic results, as well as more recent developments in this vibrant area of mathematical logic. Concrete mathematical examples are included throughout to make the concepts easier to follow. The book also contains over 200 exercises, many with solutions, making the book a useful resource for graduate students as well as researchers.

We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs. This is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive. A key result is that, in the almost simple case with socle of fixed Lie rank, apart from very specific cases, there is such a diameter bound. This is proved using recent results on the model theory of pseudofinite fields and difference fields.

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Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that R-rank(S) ≥ 2 and let Γ be a uniform lattice in G.(a) If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.(b) If CH fails, then Γ has 2 2 ω asymptotic cones up to homeomorphism.Equivalently, if µ : P(ω) → {0, 1} is the function such that µ(A) = 1 if and only if A ∈ D, then µ is a finitely additive probability measure on ω such that µ(F ) = 0 for all finite subsets F of ω. It is easily checked that if D is a nonprincipal ultrafilter and (r n ) is a bounded sequence of real numbers, then there exists a unique real number ℓ such thatfor all ε > 0. We write ℓ = lim D r n .Definition 1.3. Suppose that D is a nonprincipal ultrafilter over ω. Let (X, d)