An upper bound for van der Waerden-like numbers using k colors

Functions analogous to the van der Waerden numbers w(n, k) are considered. We replace the class of arithmetic progressions, A, by a class A′, with A ⊂A′; thus, the associated van der Waerden-like number will be smaller for si’. We consider increasing sequences of positive integers x₁,…, x_n which are either arithmetic progressions or for which there exists a polynomial φ(x) with integer coefficients satisfying φ(xi) = x_i+1, i = 1,…, n - 1. Various further restrictions are placed on the types of polynomials allowed. Upper bounds are given for the corresponding functions w′(n, k) for the general pair (n, k). A table of several new computer-generated values of these functions is provided.