Density versions of two generalizations of Schur's theorem
Given a partition of the natural numbers into finitely many cells, I. J. Schur proved that there exist x, y in some cell such that \(x+y\) is also in that cell. One of the authors [\textit{N. Hindman}, J. Comb. Theory, Ser. A 17, 1-11 (1974; Zbl 0285.05012)] has strengthened this to conclude that one of the cells contains an infinite subset, B, with the property that any finite sum of distinct elements of B is in this cell. The first result of this paper strengthens this still further. The cell in question can be chosen to have positive upper density. If \(\delta >0\) is chosen less than this upper density and if G is any finite subset of B, \(| G| =n\), let S be the set of elements from the cell for which adding any sum of distinct elements of G yields another element of the cell. Then S has upper density \(>\delta^{2\quad n}.\) Brauer's common generalization of the Schur and van der Waerden theorems states that given k, there exist x, y in some cell such that the arithmetic progression \(x,x+y,...,x+(k-1)y\) is a subset of that cell. Let Y be the set of y's in a given cell for which there is a set of x's of positive upper density with the property that \(x,x+y,...,x+(k-1)y\) are in the same cell as y. The authors' second result states that in some cell, Y has positive upper density.
scientific article; zbMATH DE number 4045751
scientific article