# Algebra Seminar talk

2010-04-23

Michael **Pinsker***Reducts of homogeneous structures with the Ramsey property*

Abstract:

For a given countably infinite *ultrahomogeneous structure S* in a
finite relational language, we are interested the problem of
classifying its *reducts*, i.e., those relational structures which have
a first-order definition in S. A conjecture of Simon Thomas from 1991
states that the number of such reducts is, up to
first-order-interdefinability, always finite. The conjecture holds for
many prominent structures, such as the dense linear order or the
random graph.

The reducts of S, factored by the equivalence of
first-order-interdefinability, correspond precisely to the *closed
supergroups of the automorphism group of S*. The problem thus is to
find these groups. This seems to be more feasible if S (or at least S
together with a suitable order) has the *Ramsey property*, i.e., if its
set of finite induced substructures is a Ramsey class: This is because
in that case, one can use Ramsey-theoretic methods in order to find
regular *patterns* in any permutation, and the group generation
process can be understood better.

In the lecture, I will show how to find such patterns. I will also explain how under this additional assumption one can prove that the number of minimal closed supergroups of the automorphism group of S is finite.

Finer classifications of reducts (e.g., up to existential or primitive positive interdefinability) require the study of other objects such as closed transformation monoids and closed clones that contain the automorphisms of S; we discuss the advantages of S having the Ramsey property for such investigations.