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) $\le$ R n (3) -- 2, between the Schur number S(n) and the Ramsey number R n (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 R n (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) $\le$ n(S(n -- 1) + 1) for all n $\ge$ 2. If true, it would yield substantially better upper bounds on the Schur numbers, e.g. S(6) $\le$ 966 conjecturally, whereas all is known so far is 536 $\le$ S(6) $\le$ 1836.