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.
2026-03-26 12:42:32.1774528952
Un-monochromatic Arithmetic Progressions
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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.