Article
The Schur degree of additive sets
Author/s | Eliahou, Shalom
Revuelta Marchena, María Pastora |
Department | Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII) |
Publication Date | 2021-05 |
Deposit Date | 2022-07-18 |
Published in |
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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 ... 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. |
Citation | Eliahou, S. y Revuelta Marchena, M.P. (2021). The Schur degree of additive sets. Discrete Mathematics, 344 (112332) |
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