On Schur properties of random subsets of integers
A celebrated theorem of Schur asserts that for each \(r>1\) and for \(n>n(r)\), if the set \(\{1,2,\dots,n\}\) is partitioned into \(r\) classes then one of them contains a solution to the equation \(x+y=z\). It is said to be a monochromatic solution. Any solution with \(x\neq y\) is called a Schur triple. A set \(A\subseteq\{1,\dots,n\}\) has the Schur property if for any 2-partition of \(A\) there is a monochromatic Schur triple. The authors prove that \(\sqrt n\) is a threshold for the Schur property showing that for every \(\omega(n)\to\infty\), almost all sets \(A\subseteq\{1,\dots,n\}\) with \(|A|>\omega(n)\sqrt n\) have the Schur property, while almost all \(A\) with \(|A|<\sqrt n/\omega(n)\) do not have this property.
scientific article; zbMATH DE number 980276