Weak Schur numbers and the search for G.W. Walker's lost partitions

  • Authors:

  • S. Eliahou;J. M. MaríN;M. P. Revuelta;M. I. Sanz

  • Affiliations:

  • Univ Lille Nord de France, F-59000 Lille, France and ULCO, LMPA J. Liouville, B.P. 699, F-62228 Calais, France and CNRS, FR 2956, France;Escuela Técnica Superior de Ingeniería de Edificación, Departamento de Matemática Aplicada I, Avenida Reina Mercedes 4, C.P. 41012 Sevilla, Spain;Escuela Técnica Superior de Ingeniería de Edificación, Departamento de Matemática Aplicada I, Avenida Reina Mercedes 4, C.P. 41012 Sevilla, Spain;Escuela Técnica Superior de Ingeniería de Edificación, Departamento de Matemática Aplicada I, Avenida Reina Mercedes 4, C.P. 41012 Sevilla, Spain

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2012

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Abstract

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.