Geodesic
On the Set of Common Differences in van der Waerden’s Theorem on Arithmetic Progressions
Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 25-36

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1999-003-9
Mots-clés : 11B25, 05D10 
Brown, Tom C.; Graham, Ronald L.; Landman, Bruce M. On the Set of Common Differences in van der Waerden’s Theorem on Arithmetic Progressions. Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 25-36. doi: 10.4153/CMB-1999-003-9
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