3-color Schur numbers

Abstract

Let $k\ge 3$ be an integer, the Schur number $S_k\left(3\right)$ is the least positive integer, such that for every 3-coloring of the integer interval $[1,S_k\left(3\right)]$ there exists a monochromatic solution to the equation $x_1+\cdots +x_k=x_{k+1}$, where $x_i$, $i=1,\dots ,k$ need not be distinct.

In 1966, a lower bound of $S_k\left(3\right)$ was established by Znám (1966). In this paper, we determine the exact formula of $S_k\left(3\right)=k^3+2k^2-2$, 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.