In this paper, we study the classification of sequences containing arbitrarily long arithmetic progressions. First, we deal with the question how the polynomial map [Formula: see text] can be extended so that it contains arbitrarily long arithmetic progressions. Under some growth conditions, we construct sequences which contain arbitrarily long arithmetic progressions. Also, we give a uniform and explicit arithmetic progression rank bound for a large class of sequences. Consequently, a dichotomy result is deduced on the finiteness of the arithmetic progression rank of certain sequences. Therefore, in this paper, we see a way to determine the finiteness of the arithmetic progression rank of various sequences satisfying some growth conditions.

Srivastava [12] showed that when the action of a finite topological group on a Tychonoff space is given, the Stone-Čech compactification of the orbit space of the action is the orbit space of the Stone-Čech compactification of the space. In this paper, we show that this statement holds also for the Hewitt realcompactification. As an application, we show that Srivastava's result can be obtained using the main theorem of the present study.

Let R be a commutative ring with identity. In this paper, for a given monotone decreasing positive sequence and an increasing sequence of subsets of R, we will define a metric on R using them. Then, we will use this kind of metric to obtain a variant of the proof of Van der Waerden's theorem by Furstenberg [3].