Generalized van der Waerden numbers (Q1086281)

A p-sequence of length n is a strictly increasing sequence of positive integers \(\{x_ 1,...,x_ n\}\) for which there exists a polynomial f with integer coefficients such that \(x_{j+1}=f(x_ j)\) for \(j=1,2,...,n-1\). Since every arithmetic progression of length n is a p- sequence of length n, the author studies the following generalization of van der Waerden numbers: p(n,k) is the least positive integer such that if \(\{\) 1,2,...,p(n,k)\(\}\) is k-colored, then there must exist a monochromatic p-sequence of length n. Several exact values of these new numbers p(n,k) are given and some upper bounds are proved. An interesting result is that if \(n\geq 4\) then \[ \lim_{m\to \infty}N(p,n,m)/N(w,n,m)\quad <\quad \infty \] where N(w,n,m) and N(p,n,m) denote the number of different arithmetic progressions and the number of different p-sequences, respectively, of length n that are contained in \(\{\) 1,2,...,m\(\}\).