Descriptive Complexity in the Finite and Infinite

forthcoming

We propose to study infinite, definable (Borel, measurable, etc.) graphs and structures and their interplay with finite ones from the perspective of descriptive complexity. Our focus will be on the following three directions: -Determining the descriptive complexity of constraint satisfaction problems (CSPs), or, equivalently, homomorphism problems on infinite domains. -Analyzing the concept of Borel hyperfiniteness of graphs and establishing new examples of non-hyperfinite graphs using methods from infinite-dimensional Ramsey theory. -Developing finitary analogues of the Borel hierarchy from descriptive set theory, constructing generalizations of the LOCAL model of distributed computing, and transferring theorems from descriptive set theory to this context. Through this research, we will achieve a deeper understanding of complexity hierarchies arising from CSPs, further advance the field of descriptive combinatorics, and transfer techniques between the finite and infinite worlds.