The Schur degree of additive sets

Abstract

Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x+y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n) ≤ Rn(3)−2, between the Schur number S(n) and the Ramsey number Rn(3) = R(3, . . . , 3), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of G. Recursive upper bounds are known for Rn(3) but not for S(n) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our study of the Schur degree leads us to conjecture S(n) ≤ n(S(n − 1) + 1) for all n ≥ 2. If true, it would yield substantially better upper bounds on the Schur numbers, e.g. S(6) ≤ 966 conjecturally, whereas all is known so far is 536 ≤ S(6) ≤ 1836.