On sets of integers with the Schur property (Q1821785)

The authors investigate additional properties of theorems due to Rado's and Schur's (Rado's theorem being the statement that for every r coloring of the positive integers there exist k integers such that all their finite sums - without repetition - are colored the same; Schur's result is \(k=2)\). Using standard constructions sparse versions are established, viz., a graph theoretic result of the authors implies a strong sparse version of Schur's theorem; the probabilistic method is used to derive a weak sparse version of Rado's theorem.