Pseudo-arithmetic sets and Ramsey theory

Given a set A of natural numbers, let d(A)={(y−x) | x<y∈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∈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)⩾614. We prove a divisible Schur theorem: for any c>0 there is a number s=Sd(c), so that for any c-coloring χ:[s]→[c], there is a monochromatic set {x,y,x+y} with x | y.