On the Set of Common Differences in van der Waerden’s Theorem on Arithmetic Progressions

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

Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 25-36

Résumé

Analogues of van der Waerden’s theorem on arithmetic progressions are considered where the family of all arithmetic progressions, $\text{AP}$ , is replaced by some subfamily of $\text{AP}$ . 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 $n={w}'\text{(}k,r)$ 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 $\{{{a}_{n}}\,:\,n\,=\,1,\,2,\ldots \}$ can have $\underset{n\to \infty }{\mathop{\lim \,\inf }}\,\,\frac{{{a}_{n+1}}}{{{a}_{n}}}\,>\,1$ . 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.

Détail

DOI : 10.4153/CMB-1999-003-9

Mots-clés : 11B25, 05D10