Un-monochromatic Arithmetic Progressions Prove that the set $\\{1,2,3,...,2008\\}$ can be colored with two colors such that any $18$ term arithmetic progression in this set is not monochromatic.

Randomly colour the members of the set black and white, independently with probabilities $1/2$ and $1/2$.
The probability that any given $18$-term a.p. in the set is monochromatic is $2^{-17}$. There are $117587$ such a.p.'s, and this is less than $2^{17}$. Thus the expected number of monochromatic a.p.'s is less than one, which means that it must be possible to have no monochromatic a.p.'s.