  A set $A$ of integers is said to be 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 $|A|> \lceil\tfrac{4n}{5}\rceil$, 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
$\omega(n^{1/3})$ random integers suffices, noting that this is optimal for
sets $A$ with $|A|\leq \lceil\tfrac{n}{2}\rceil$. We close the gap between
these two results by showing that if $A\subseteq [n]$ with
$|A|=\lceil\tfrac{n}{2}\rceil+t<\lceil\tfrac{4n}{5}\rceil$, then adding
$\omega(\min\{n^{1/3},nt^{-1}\})$ 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.
