Erdős and Hajnal showed that graphs satisfying any fixed hereditary property contain much larger cliques or independent sets than what is guaranteed by (the quantitative form of) Ramsey's theorem. We start with a whirlwind tour of the history of this observation, and then we present some new results for ordered graphs, that is, for graphs with a linear ordering on their vertex sets.
The talk will introduce, hopefully at a basic level, the meaning and analysis of spaces with Ricci curvature bounds. We will discuss the process of limiting spaces with such bounds, and studying the singularities on these limits. The singularities come with a variety of natural structure which have been proven in the last few years, from dimension bounds to rectifiable structure, which is (measure-theoretically) a manifold structure on the singular set. If time permits we will discuss some recent work involving the topological structure of boundaries of such spaces.