Monochromatic Schur Triples in Randomly Perturbed Dense Sets of Integers

Given a dense subset A of the first n positive integers, we provide a short proof showing that for p = omega(n(-2/3)), the so-called randomly perturbed set A boolean OR[n](p) a.a.s. has the property that any 2-coloring of it has a monochromatic Schur triple, i.e., a triple of the form (a, b, a + b). This result is optimal since there are dense sets A, for which A boolean OR [n](p) does not possess this property for p = o(n(-2/3)).