Schur properties of randomly perturbed sets

Abstract

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⊆[n] to ensure with high probability that the resulting set is Schur? Hu showed in 1980 that when |A|>⌈4n5⌉, 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⊆[n], adding ω(n1/3) random integers suffices, noting that this is optimal for sets A with |A|≤⌈n2⌉. We close the gap between these two results by showing that if A⊆[n] with |A|=⌈n2⌉+t<⌈4n5⌉, then adding ω(min{n1/3,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.