Coloring the integers

The answer is 9. This a consequence of van der Waerden's theorem, which states: if $N = A_1 ∪ ... ∪ A_k$ is a partition of the sets of N, then one of the sets $A_1, ..., A_k$ contains finite arithmetic progressions of arbitrary length. Given positive integers $m$ and $l$ , there is a positive integer $m$ such that if the set {1, 2, ..., m} is divided into $k$ classes, at least one class contains $l+1$ numbers which form an arithmetic progression. We denote the least number $m$ possessing this property by $W(k, l)$ and call it the van der Waerden number. The question is equivalent to asking for the van der Waerden number $W(2, 3)$ , which is 9.