On the set of common differences in van der Waerden's theorem on arithmetic progressions

Tom C. Brown, Ronald L. Graham, Bruce M. Landman

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

Analogues of van der Waerden's theorem on arithmetic progressions are considered where the family of all arithmetic progressions, AP, is replaced by some subfamily of AP. Specifically, we want to know for which sets A, of positive integers, the following statement holds: for all positive integers r and k, there exists a positive integer n = w′(k, r) such that for every r-coloring of [1, n] there exists a monochromatic k-term arithmetic progression whose common difference belongs to A. We will call any subset of the positive integers that has the above property large. A set having this property for a specific fixed r will be called r-large. We give some necessary conditions for a set to be large, including the fact that every large set must contain an infinite number of multiples of each positive integer. Also, no large set {an : n = 1,2,...}can have n→∞lim inf an+1/an > 1. Sufficient conditions for a set to be large are also given. We show that any set containing n-cubes for arbitrarily large n, is a large set. Results involving the connection between the notions of "large" and "2-large" are given. Several open questions and a conjecture are presented.

Original language English (US) 25-36 12 Canadian Mathematical Bulletin 42 1 https://doi.org/10.4153/CMB-1999-003-9 Published - Jan 1 1999 Yes

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Common difference
Arithmetic sequence
Integer
Large Set
Theorem
N-cube
Colouring
Analogue
Necessary Conditions
Subset
Sufficient Conditions
Term

ASJC Scopus subject areas

• Mathematics(all)

Cite this

On the set of common differences in van der Waerden's theorem on arithmetic progressions. / Brown, Tom C.; Graham, Ronald L.; Landman, Bruce M.

In: Canadian Mathematical Bulletin, Vol. 42, No. 1, 01.01.1999, p. 25-36.

Research output: Contribution to journalArticle

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