From Hindman to Gowers

Abstract

The seminal results of Schur (Schur’s Lemma) and Hilbert (Hilbert Cube Lemma) eventually led to the development of a whole research area at the interface between Ramsey theory and additive combinatorics. In this context, one studies which additive combinatorial configurations in $\mathbb {N}$ are partition regular, i.e. they can be found within a color of each finite coloring of $\mathbb {N}$. While van der Waerden’s theorem on arithmetic progressions (discussed in the previous chapter) is the most famous early result in this area, several other additive configurations were later shown to be partition regular. Among these there are sets of finite sums of a finite sequence, which is the content of Folkman’s theorem.