Pseudo-arithmetic sets and Ramsey theory

摘要

Given a set A of natural numbers, let d(A) = {(y - x) x < y is an element of A} and let m(A) min(d(A)). The set A is said to be pseudo-arithmetic if m(A) (x - y) for all x, y is an element of A. We prove a pseudo-arithmetic Ramsey theorem: for any c, k, n > 0 there is a number p = P(n; c, k), so that for any c-coloring f :[p](k) --> [c], there is a pseudo-arithmetic set A with A = n and f constant on [A](k). We prove that P(3, 2, 2) = 13, and show that P(3, 3, 2) greater than or equal to 614. We prove a divisible Schur theorem: for any c > 0 there is a number s - S,1(c), so that for any c-coloring chi : [s] - [c], there is a monochromatic set {x, y, x + y} with x y. (C) 2004 Elsevier Inc. All rights reserved.