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

AbstractNumbers similar to those of van der Waerden are examined. We consider increasing sequences of positive integers {x1, x2, …, xn} either that form an arithmetic sequence or for which there exists a polynomial f(x) = Σi = 0n − 2 aixi with ai ϵ Z, an − 2 > 0, and xj + 1 = f(xj). 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