Schur properties of randomly perturbed sets

Abstract

A set $A$ of integers is said to be \emph{Schur} if any two-colouring of $A$ results in monochromatic $x,y$ and $z$ with $x+y=z$. We study the following problem: how many random integers from $[n]$ need to be added to some $A\subseteq [n]$ to ensure with high probability that the resulting set is Schur? Hu showed in 1980 that when $|A|> \lceil 4n/5 \rceil$, no random integers are needed, as $A$ is already guaranteed to be Schur. Recently, Aigner-Horev and Person showed that for any dense set of integers $A\subseteq [n]$, adding $ω(n^{1/3})$ random integers suffices, noting that this is optimal for sets $A$ with $|A|\leq \lceil n/2 \rceil$. We close the gap between these two results by showing that if $A\subseteq [n]$ with $|A|=\lceil n/2 \rceil +t<\lceil 4n/5 \rceil$, then adding $ω(\min\{n^{1/3},nt^{-1}\})$ random integers will with high probability result in a set that is Schur. Our result is optimal for all $t$, and we further provide a stability result showing that one needs far fewer random integers when $A$ is not close in structure to the extremal examples. We also initiate the study of perturbing sparse sets of integers $A$ by using algorithmic arguments and the theory of hypergraph containers to provide nontrivial upper and lower bounds.