Abstract: In this paper,we translate the van der Waerden number problem into the van der Waerden number problem on circle,and we find the equivalent form Ｓｐ（ｋ）　of its inequality.At the same time,we also obtain formula  of local system of inequality,especially we obtain the lower and upper bounds of Ｓｐ（ｋ）.

Key words: van der Waerden number　Ｗ（ｎ, ｎ）；Lucas-Fibonacci sequence;lower bound;upper bound