Three-in-one in Ramsey theory

Abstract: There are three fundamental theorems in Ramsey theory: Ramsey's theorem, van der Waerden's theorem, and Hindman's theorem. Milliken-Taylor proved a result that simultaneously generalizes Ramsey's theorem and Hindman's theorem. Later, Bergelson--Hindman and Samet--Tsaban established two distinct theorems, each providing a simultaneous generalization of Ramsey's theorem and van der Waerden's theorem in two different ways. Using a newly developed method of iterated extensions, we prove--pending verification--a theorem that simultaneously generalizes all three classical results--Ramsey's theorem, Hindman's theorem, and van der Waerden's theorem. Moreover, our theorem subsumes both the Bergelson--Hindman and the Samet--Tsaban generalizations. Joint work with Mauro Di Nasso.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)