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

Abstract

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.