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 {sl randomly perturbed} set $A cup [n]_p$ a.a.s. has the property that any $2$-colouring 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 $Acup [n]_p$ does not possess this property for $p=o(n^{-2/3})$.