ABSTRACT
For integers k, n, c with k, n ⩾ 1, and c ⩾ 0, the n-color weak Rado number 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
with xi ≠ xj, when i ≠ j. If no such N exists, then
is defined as infinite.
In this paper, we determine the exact value of some of these numbers for n = 2 and n = 3, namely ,
for all c ⩾ 0 and
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.
Acknowledgment
The authors wish to thank an anonymous referee for your very careful reading of this paper and your sharp comments correcting of a few dubious points.