Artículo
3-color Schur numbers
Autor/es | Boza Prieto, Luis
Marín Sánchez, Juan Manuel Revuelta Marchena, María Pastora Sanz Domínguez, María Isabel |
Departamento | Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII) |
Fecha de publicación | 2019 |
Fecha de depósito | 2022-07-29 |
Publicado en |
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Resumen | Let k ≥ 3 be an integer, the Schur number Sk(3) is the least positive integer, such that for
every 3-coloring of the integer interval [1, Sk(3)] there exists a monochromatic solution to
the equation x1+ · · · + xk= xk+1, ... Let k ≥ 3 be an integer, the Schur number Sk(3) is the least positive integer, such that for every 3-coloring of the integer interval [1, Sk(3)] there exists a monochromatic solution to the equation x1+ · · · + xk= xk+1, where xi , i = 1, . . . , k need not be distinct. In 1966, a lower bound of Sk(3) was established by Znám (1966). In this paper, we determine the exact formula of Sk(3) = k 3 + 2k 2 − 2, finding an upper bound which coincides with the lower bound given by Znám (1966). This is shown in two different ways: in the first instance, by the exhaustive development of all possible cases and in the second instance translating the problem into a Boolean satisfiability problem, which can be handled by a SAT solver. |
Cita | Boza Prieto, L., Marín Sánchez, J.M., Revuelta Marchena, M.P. y Sanz Domínguez, M.I. (2019). 3-color Schur numbers. Discrete Applied Mathematics, 263, 59-68. |
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