One nice feature of Fraisse limits for finite languages is that there is an associated zero-one law for the finite structures in the corresponding Fraisse class. We show that for Fraisse classes over infinite relational languages that locally are finitary in a suitable sense, there are also corresponding zero-one laws under an appropriate choice of probability measure. Examples of such Fraisse classes are the classes of simplicial complexes and of hypergraphs. We contrast our zero-one law with the known zero-one law of Blass and Harary.

In this paper, we study a class of initial segments of models of arithmetic, called thin initial segments. Using indicator theory of Paris and Kirby, we can find an unprovable version of thin set theorem.

A structure is small if it has countably many pure n-types for each integer n. Such structures arise when one whishes to count the number of pairwise non-isomorphic countable models of a given theory. Weakly small structures have been introduced by Belegradek to give a common generalisation of small and minimal structures. A structure is weakly small if for every finite tuple A coming from this structure, there are countably many 1-types over A. We shall show that weakly small algebraic structures behave very much like omega-stable ones, at least locally: in a weakly small group, subgroups which are definable with parameters in a finitely generated algebraic closure satisfy local descending chain conditions. An infinite weakly small group has an infinite abelian subgroup. A nilpotent small group is the central product of a definable divisible group with one of bounded exponent. Every weakly small division ring of positive characteristic is locally finite dimensional over its centre. The Jacobson radical of a weakly small ring is nil, locally nilpotent. Every weakly small division ring is locally modulo its radical the product of finitely many matrix rings over division rings.

Continuum size structures occur naturally in analysis, algebra, and other areas. Examples are the additive group of real numbers, and the ring of continuous functions on the unit interval. How about effectiveness constraints on their presentations? A reasonable approach is to require that domain and relations are Borel. Most examples of structures from the areas above have presentations of this kind.

Borel structures were introduced by H. Friedman in 1978. He proved that each countable theory has a Borel model of size the continuum. His main interest, however, was in quantifiers such as ``there exists a co-meager set of x's such that ...'' Some results were obtained till the late 1990s, for instance results on Borel partial orders by Harrington and Shelah.

The subject was revived by work with Hjorth, Khoussainov, and Montalban (LICS 2008). We were primarily interested in presentations via Büchi automata (which process infinite strings of symbols). For instance, the additive group of reals is Büchi presentable; so is the Boolean algebra B of sets of natural numbers modulo finite differences. All Büchi presentations are Borel. In the LICS paper it was shown that some Büchi presentable structure close to the Boolean algebra B does not have an injective Borel presentation (where each element is represented uniquely). This answered an open question from the theory of automatic structures. It is still unknown whether B itself has an injective presentation.

So far the language was assumed to be countable. Hjorth and I considered the more general case where the language is uncountable but Borel (for instance, the language of a vector space over the reals). In a recent JSL paper we showed that the completeness theorem fails for Borel structures in this wider sense: some complete Borel theory has no Borel model.

I will end the talk with open questions. Woodin asked whether each Borel Scott set is the standard system of a Borel model of PA. Further, does every Borel field Borel embed into a Borel algebraically closed field? If not, this would yield an alternative proof of the result with Hjorth.

Given a graph H, we say that a graph G is H-free if H does not embed into G as a subgraph, induced or otherwise. Cherlin, Shelah and Shi (1999) have shown that for any fixed, finite, connected graph H, the theory of the existentially complete H-free graphs is complete and model complete; further, they give an elegant criterion under which this theory is omega-categorical. The question then arises: given a finite connected graph H, where does the theory of the existentially complete H-free graphs lie within Saharon Shelah's classification spectrum, which is a taxonomy for complete first order theories based on certain syntactic properties? We are especially interested in the region of Shelah’s classification consisting of theories that have the so-called n-strong order properties (SOP_n), n >2. Among these, theories with SOP₃ but not SOP₄ are considered the most tractable. I will provide a general condition, related to the Cherlin, Shelah and Shi criterion for omega-categoricity, for the failure of SOP₄, and use this to give an example of an infinite family of graphs H for which the theories of the existentially complete H-free graphs all possess SOP₃ but not SOP₄. I will also discuss partial results and open questions concerning the classification of the theory of existentially complete H-free graphs for an arbitrary finite connected graph H. All definitions will be given.

A directed set D is said to be Tukey reducible to another directed set E, written D <=_T E, if there is a function f: D -> E which maps unbounded subsets of D to unbounded subsets of E. We say D and E are Tukey equivalent if D <=_T E and E <=_T D. The notion of Tukey equivalence tries to capture the idea that two directed posets "look cofinally the same", or have the same "cofinal type". As such, it provides a device for a "rough classification" of directed sets based upon their "cofinal type", as opposed to an exact classification based on their isomorphism type. This notion has recently received a lot of attention in various contexts in set theory. In joint work with Todorcevic, I have investigated the Tukey theory of ultrafilters on the natural numbers, which can naturally be viewed as directed sets under reverse containment. In the case of ultrafilters, Tukey reducibility is coarser than the well studied Rudin-Keisler reducibility (RK reducibility). I will present some recent progress on the Tukey theory of ultrafilters, focusing on the question "under what conditions is Tukey reducibility actually equivalent to RK reducibility?".

We completely characterize all definable linear orders in sufficiently rich o-minimal structures. Let M be an o-minimal structure expanding a field, for instance the real field. Let (P,<_p) be any definable linear order in M. Then (P,<_p) embeds definably in (Mⁿ+1},<_l, where <l is the lexicographic order and n is the o-minimal dimension of P. This improves a result of A. Onshuus and C. Steinhorn in the case that M is o-minimal expanding a field.

Kaplansky classes emerged in the context of Enoch's’ solution of the Flat Cover Conjecture. Their connection to abstract model theory goes back to [3]: a class C of roots of Ext is a Kaplansky class closed under direct limits, iff the pair (C,<=) is an abstract elementary class (AEC) in the sense of Shelah. A question was raised whether this AEC has finite character. We give a positive answer in case C = ^⊥C' for a class of pure–injective modules C0. This yields a positive answer for all AECs of roots of Ext over any right noetherian right hereditary ring R.

If (C;<=) is an AEC of roots of Ext then C is known to be a covering class. However, Kaplansky classes need not even be precovering in general: We prove that the class D of all aleph₁-–projective modules is a Kaplansky class for any ring R, but it fails to be precovering in case R is not right perfect, the class ^⊥(D^⊥) equals the class of all flat modules and consists of modules of projective dimension <= 1. Assuming the Singular Cardinal Hypothesis, we prove that D is not precovering for each countable non–right perfect ringR.

There exists a natural way to extend the operation of any groupoid (in fact, any universal algebra) to ultrafilters; the extended operation is right topological in the standard compact Hausdorff topology on the set of ultrafilters; the extensions of semigroups are semigroups. Semigroups of ultrafilters are used to obtain various deep results of number theory, algebra, dynamics, etc. The main tool is idempotent ultrafilters. They exist by a general theorem establishing the existence of idempotents in compact Hausdorff right topological semigroups.

Expanding this technique to non-associative groupoids, we isolate a class of formulas such that any satisfying them compact Hausdorff right topological groupoid has an idempotent, and a class of formulas that are stable under passing from a given groupoid to the groupoid of ultrafilters. If a formula belongs to both classes (like associativity), any satisfying it groupoid carries an idempotent ultrafilter. Results on semigroups following from the existence of idempotent ultrafilters (like Hindman's Finite Sums Theorem) remain true for such groupoids.

Another generalization concerns infinitary analogs of these results. The main obstacle here is that non-principal idempotent ultrafilters cannot be σ-additive. We define ultrafilters with two weaker properties (ultrafilters close to κ-additive subgroupoids and κ-additive ultrafilters close to subgroupoids) and show that their existence suffices to obtain desired infinitary theorems.

Let (Ω, A) be a measurable space with A closed under Souslin operations and Y be a Polish space with B_Y denoting its Borel σ-algebra. Suppose B in A⊗B_Y (the product σ-algebra) with sections B_ω = {y in Y:(ω,y) in B} uncountable for each ω \in Ω. Then there is a map Q:Ω × B_Y -> [0,1] satisfying the following conditions:

1. For each ω in Ω, Q(ω, .) is a continuous probability with (topological) support contained in B_ω.
2. For each Borel subset B of Y, ω -> Q(ω, B) is A-measurable.