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

AbstractA 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 ‖P‖ ⩽ h(a) and ‖P‖ ⩽ 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 ‖A‖ > f(d), where d is the maximum of the differences between consecutive elements of A