On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1

Abstract

For integers k, n, c with k, n ≥ 1, and c ≥ 0, the n-color weak Rado number WRk (n, c) is defined as the least integer N, if it exists, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution x1 ,..., xk, xk+1 in that interval to the equation x1 + x2 +···+ xk + c = xk+1 , with xi = xj , when i = j. If no such N exists, then WRk (n, c) is defined as infinite. In this paper, we determine the exact value of some of these numbers for n = 2 and n = 3, namely WR3 (2, c) = 5c + 24, WR4(2, c) = 6c + 52 for all c ≥ 0 and WR2 (3, c) = 13c + 22 for all c > 0. Our method consists in translating the problem into a Boolean satisfiability problem, which can then be handled by a SAT solver or by backtrack programming in the language C.