| Résume | First introduced by Bourgain, Figiel and Milman in the context of the Ribe Program, Ramsey-type theorems for metric spaces state that compact metric spaces contain ``large" subsets which are approximately ultrametric. They have since found applications in Metric Geometry, Online Algorithms, Approximation Algorithms, Data Structures, Probability, Descriptive Set Theory, and Geometric Measure Theory. We will discuss the ``ultrametric skeleton" theorem behind most Ramsey-type theorems for metric spaces, and a recent regular form of the theorem. Time permitting, we will prove of a special case of the theorem for doubling spaces. |
