Abstract: In this paper we seek a good way to improve the up-bound of the van der Waerden number. We translate the problem of van der Waerden number on circle into a linear system of inequalities in a matrix form. We study the matrix form to find a good up-bound of the van der Waerden number on circle, and find two particular results. When p=nr±1, we obtain an expression of x and b_k.

Key words: W_h(n,n), matrix, van der Waerden number