Finitely Bounded Homogeneous Structures

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Homogeneous structures play an important role in model theory, where they provide a rich source of examples and counterexamples. They have a large automorphism group and can be studied using the theory of infinite permutation groups. They naturally appear in several areas of mathematics. Many of the important examples of homogeneous structures are *finitely bounded*, that is, can be described by finitely many forbidden finite substructures. In this way we store and manipulate homogeneous structures computationally, and many fundamental questions about homogenous structures can be posed as algorithmic questions. Indeed, homogeneous structures have applications in theoretical computer science, for example for the study of the computational complexity of constraint satisfaction problems (CSPs), for the studying normal representations of relation algebras, or computation with definable structures. A relatively young and powerful tool in the study of homogeneous structures is structural Ramsey theory. Ramsey theorems for homogeneous structures have applications in the mentioned application areas, but also in topological dynamics, for example for verifying extreme amenability, amenability, and unique ergodicity of topological groups. One of the goals of the project is to answer fundamental open questions about homogeneous structures for large classes, e.g., classes obtained by imposing further model-theoretic assumptions or restrictions on the orbit growth rate. In this way we hope to obtain insights for the mentioned application areas as well.