On van der Waerden's theorem and the theorem of Paris and Harrington

A 2-coloring of the non-negative integers and a function h are given such that if P is any monochromatic arithmetic progression with first term a and common difference d then 6P6 ? h(a) and 6P6 ? h(d). In contrast to this the following result is noted. For any k, f there is n = n(k, f) such that whenever n is k-colored there is a monochromatic subset A of n with 6A6 > f(d), where d is the maximum of the differences between consecutive elements of A.