Ramsey's coheirs
Eugenio Colla, Domenico Zambella
TL;DR
This paper develops a model-theoretic framework based on coheirs to provide succinct proofs of key Ramsey-type results, replacing the Stone-Čech semigroup with a saturated monster model of a semigroup $G$. By constructing coheir sequences that are $M$-indiscernible, the authors derive Ramsey's theorem and extend it to Hindman and Hales-Jewett-type results, including Carlson’s and Gowers’s partition theorems as consequences of an idempotent-orbit structure in semigroups. The approach yields algebraic and combinatorial versions of the theorems, emphasizes stationarity and independence notions, and demonstrates how model-theoretic methods can illuminate classical combinatorial phenomena. The work offers an accessible alternative to ultrafilter methods and strengthens connections between Ramsey theory and model theory through a unified coheir-based toolkit.
Abstract
We use the model theoretic notion of coheir to give short proofs of old and new theorems in Ramsey Theory. As an illustration we start from Ramsey's theorem itself. Then we prove Hindman's theorem and the Hales-Jewett theorem. Finally, we prove two Ramsey theoretic principles that have among their consequences partition theorems of Carlson and of Gowers.