On Schur Properties of Random Subsets of Integers Original Research Article

A classic result of I. Schur [9] asserts that for everyrgreater-or-equal, slanted2 and fornsufficiently large, if the set [n]={1, 2, …, n} is partitioned intorclasses, then at least one of the classes contains a solution to the equationx+y=z. Any such solution withx≠ywill be called aSchur triple. Let us say thatAsubset of or equal to[n] has theSchur propertyif for every partition (or 2-coloring) ofA=Runion or logical sumB(forredandblue), there must always be formed somemonochromaticSchur triple, i.e., belonging entirely to eitherRorB. Our goal here is to show thatn1/2is a threshold for the Schur property in the following sense: for anyω=ω(n)→∞, almost all setsAsubset of or equal to[n] with Aωn1/2do.