Ramsey functions related to the van der waerden numbers

Ramsey functions similar to the van der Waerden numbers w(n) are studied. If A' is a class of sequences which includes the n-term arithmetic progressions, then we define w'(n) to be the least positive integer guaranteeing that if {1,2,...,w'(n)} is 2-colored, then there exists a monochromatic member of A'. We consider increasing sequences of positive integers {x₁,...,x_n} which are either arithmetic progressions or for which there exists a polynomial p(x) with integer coefficients satisfying p(x_i) = x_i+1. Various further restrictions are placed on the types of polynomials allowed. Upper bounds are given for the corresponding functions w'(n). In addition, it is shown that the existence of somewhat stronger bounds on w'(n) would imply similar bounds for w(n).