Schur properties of randomly perturbed sets

TL;DR

This paper investigates how many random integers must be added to a subset of [n] to ensure it becomes Schur, bridging previous results and establishing optimal thresholds with stability and algorithmic insights.

Contribution

It extends prior work by determining the minimal random additions needed for sets of intermediate size to become Schur, including stability results and bounds for sparse sets.

Findings

Adding ((n^{1/3}), nt^{-1}) random integers suffices for certain set sizes.

The results are proven to be optimal across all parameters.

Stability analysis shows fewer random integers are needed when sets are structurally distant from extremal examples.

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 $A\subseteq [n]$ 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.