3-color Schur numbers

• Published in: Discrete Applied Mathematics Vol. 263; pp. 59 - 68
• Main Authors: Boza, L., Marín, J.M., Revuelta, M.P., Sanz, M.I.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 30.06.2019; Elsevier BV
• Subjects: [formula omitted]-coloring; Boolean algebra; Coloring; Integers; Lower bounds; Monochromatic solution; Numbers; Schur numbers; Sum-free sets; Upper bounds; Monochromatic solution; Sum-free sets; n-coloring; Schur numbers
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2018.06.030

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)=k3+2k2−2, 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.