Andre Nies. *The University of Auckland. Auckland, New Zealand.*

**Borel Structures**
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.

Rehana Patel. *Harvard University. Cambridge MA, United States of America.*

**Classifying Theories of Graphs with a Forbidden Subgraph**
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 SOP_{3} but not SOP_{4} 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
SOP_{4}, 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 SOP_{3} but not SOP_{4}. 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.

Dilip Raghavan. *University of Toronto. Toronto ON, Canada.*

**Cofinal types
of ultrafilters**
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?".

Janak Ramakrishnan. *Université Claude Bernard. Lyon, France.*

**Definable
linear orders definably embed into lexicographic orders in o-minimal
structures**
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 (*M*^{n}+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.

Jan Saroch. *Charles University. Prague, Czech Republic*.

**Kaplansky classes,
finite character and ***aleph*_{1} projectivity
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 *aleph*_{1}-–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 ring*R*.

Denis I. Saveliev. *Moscow State University. Moscow, Russia*.

**Groupoids of
ultrafilters**
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.