Weak Schur numbers and the search for G.W. Walker’s lost partitions
A set A of integers is weakly sum-free if it contains no three distinct elements x, y, z such that x + y = z. Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5) = 196, without pr...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2012 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/136600 |
| Acceso en línea: | https://hdl.handle.net/11441/136600 https://doi.org/0.1016/j.camwa.2011.11.006 |
| Access Level: | acceso abierto |
| Palabra clave: | Schur numbers Sum-free sets Weakly sum-free sets Boolean variables SAT problem SAT-solvers |
| Sumario: | A set A of integers is weakly sum-free if it contains no three distinct elements x, y, z such that x + y = z. Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5) = 196, without proof. Here we show WS(5) ≥ 196, by constructing a partition of {1, . . . , 196} of the required type. It remains as an open problem to prove the equality. With an analogous construction for k = 6, we obtain WS(6) ≥ 572. Our approach involves translating the construction problem into a Boolean satisfiability problem, which can then be handled by a SAT solver. |
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