Exact value of 3 color weak Rado number

Abstract

For integers k, n, c with k, n ≥ 1 and c ≥ 0, the n color weak Rado number W Rk(n, c) is defined as the least integer N, if it exists, such that for every n coloring of the set {1, 2, ..., N}, there exists a monochromatic solution in that set to the equation x1 + x2 + ... + xk + c = xk+1, such that xi = xj when i = j. If no such N exists, then W Rk(n, c) is defined as infinite. In this work, we consider the main issue regarding the 3 color weak Rado number for the equation x1 + x2 + c = x3 and the exact value of the W R2(3, c) = 13c + 22 is established.