Some new bounds and values for van der Waerden-like numbers

Numbers similar to those of van der Waerden are studied. We consider increasing sequences of positive integers {x₁, x₂,..., x_n} that either form an arithmetic progression or for which there exists a polynomial f with integer coefficients and degree exactly n - 2, and x_j+1 =f(x_j). We denote by q(n, k) the least positive integer such that if {1, 2,..., q(n, k)} is partitioned into k classes, then some class must contain a sequence of the type just described. Upper bounds are obtained for q(n, 3), q(n, 4), q(3, k), and q(4, k). A table of several values is also given.