Algebra Seminar talk
Reducts of homogeneous structures with the Ramsey property
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.