Abstract: Model theory is concerned with the relationship between sets of formulas (theories) and mathematical models (e.g. groups, vector spaces, graphs, ordered sets). One of its main themes is the description and classification of the models of a theory T. A more concrete question, which nevertheless yields insight into the classification problem, is determining the number of models of T of a given cardinality $\kappa$, up to isomorphism. Further interesting

questions are obtained when the set of "allowed" isomorphisms is restricted and when the role of isomorphism is replaced with similar natural notions, such as elementary embeddability. There is a natural embedding of the set of models with domain $\kappa$ into the $\kappa$-Baire space (the set of functions from $\kappa$ to $\kappa$ endowed with the Baire topology). This allows us to study models via investigating topological properties of the $\kappa$-Baire space and thereby obtain results pertaining to the problems described above.

In this lecture, we present a dichotomy theorem for certain binary relations of the $\kappa$-Baire space, which holds under an additional set theoretic hypothesis. We have as a corollary a dichotomy related to the number of $\kappa$-sized models

of a theory T up to isomorphism by the functions in "nice" subsets of the $\kappa$-Baire space. The role of isomorphism can be replaced by embeddability, elementary embeddability or similar notions. These results are joint with Jouko Väänänen. All related concepts will be explained in detail.