Date: Thursday, April 25, 2019
Location: Rockefeller Hall 312
Title: Arithmetic Regularity with Forbidden Bipartite Configurations
A recurring phenomenon in combinatorics is that families of finite graphs, which omit some fixed graph as an induced subgraph, satisfy stronger structural properties than arbitrary finite graphs. A well-known manifestation of this theme is the result of Erdos and Hajnal showing that, for any fixed finite graph H, an H-free graph contains a much larger complete or independent set than a “random” finite graph. In this talk, I will consider analogous questions in the setting of finite groups. Specifically, suppose A is a subset of a finite group G, such that the bipartite graph on G induced by x + y ∈ A omits some previously fixed bipartite graph as an induced subgraph. What can be said about the algebraic structure of A? The answer to this question will draw from the notion of arithmetic regularity, which is a group theoretic analogue of Szemeredi regularity for graphs, first developed by Ben Green. The results presented in this talk combine tools from model theory, additive combinatorics, and the structure of compact topological groups. Joint with A. Pillay and C. Terry.