Problems and new results on the algebra of \(\beta\mathbb{N}\) and its application to Ramsey theory (Q2772761)

The author of the article under review presents a kind of survey on the algebraic structure of \(\beta {\mathbb N}\) and \(\beta{\mathbb Z}\). A number of important new results have been proved in this area after \textit{N. Hindman} and \textit{D. Strauss}' book ``Algebra in the Stone-Čech compactification -- theory and applications'' [Walter de Gruyter \(\&\) Co., Berlin (1998; Zbl 0918.22001)] was published. The article contains proofs of two of them, a new algebraic result about sums of idempotents obtained by Zelenyuk (Section 2) and a new algebraic proof of a Ramsey theoretic result about polynomial progressions by Bergelson and Leibman (Section 3).NEWLINENEWLINENEWLINEIn Section 4, Neil Hindman discusses several open problems concerning the algebraic structure of \(\beta{\mathbb N}\). The relations between some open problems in Ramsey theory with those in the algebra of \(\beta{\mathbb N}\) are considered in Section 5 of the article.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00067].