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 $W{R}_k(n,c)$ 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, …, x_k, x_{k + 1} in that interval to the equation $$\begin{array}{c}\hfill x_1+x_2+\cdots +x_k+c=x_{k+1},\end{array}$$with x_i ≠ x_j, when i ≠ j. If no such N exists, then $W{R}_k(n,c)$ is defined as infinite.

In this paper, we determine the exact value of some of these numbers for n = 2 and n = 3, namely $W{R}_3(2,c)=5c+24$, $W{R}_4(2,c)=6c+52$ for all c ⩾ 0 and $W{R}_2(3,c)=13c+22$ 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.

KEYWORDS:

• Schur numbers
• sum-free sets
• weak Schur numbers
• weakly sum-free sets
• Rado numbers
• weak Rado numbers

Mathematics Subject Classification:

• 05C55
• 05D10
• 05-04
• 05A17

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.