On the n-Color Weak Rado Numbers for the Equation x1+x2+⋯+xk+c=xk+1
Luis B. Boza, J. M. MARIN, M.P. Revuelta, M.I. Sanz · Experimental Mathematics · 2017
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.