On the existence of a reasonable upper bound for the van der Waerden numbers

Numbers similar to those of van der Waerden are examined. We consider increasing sequences of positive integers {x₁, x₂, ..., x_n} either that form an arithmetic sequence or for which there exists a polynomial f(x) = Σ_{i = 0}^{n - 2} a_ixⁱ with a_i ε{lunate} Z, a_{n - 2} > 0, and x_{j + 1} = f(x_j). We denote by q(n) the least positive integer such that if {1, 2, ..., q(n)} is 2-colored, then there exists a monochromatic sequence of the type just described. We give an upper bound for q(n), as well as values of q(n) for n ≤ 5. A stronger upper bound for q(n) is conjectured and is shown to imply the existence of a similar bound on the nth van der Waerden number.