On Schur Properties of Random Subsets of Integers

A classic result of I. Schur [9] asserts that for every r 2 and for n sufficiently large, if the set [n]=[1, 2, ..., n] is partitioned into r classes, then at least one of the classes contains a solution to the equation x+ y=z. Any such solution with x{y will be called a Schur triple. Let us say that A [n] has the Schur property if for every partition (or 2-coloring) of A=R _ B (for red and blue), there must always be formed some monochromatic Schur triple, i.e., belonging entirely to either R or B. Our goal here is to show that n 2 is a threshold for the Schur property in the following sense: for any |=|(n) , almost all sets A [n] with |A||n 2 do.