Hypernatural numbers in arithmetic Ramsey theory

Abstract

The hypernatural numbers $\mathbb{N}$ of nonstandard analysis have recently proven to be an effective tool in arithmetic Ramsey theory. After introducing the fundamental ``nonstandard" notions, we present several examples to illustrate the use of this technique in practice. In particular, we provide brief nonstandard proofs of some recent results concerning the partition regularity of certain families of infinite configurations. We also present a new theorem about monochromatic sums and products.

To appear in the volume "Perspectives on Ergodic Theory and Its Interactions - In Honor of Vitaly Bergelson."