# Arithmetically progressive

Discrete Mathematics Level pending

Find the smallest positive integer $$N$$ with the following property. For any subset $$S$$ of the set $$\{1,2,...,N\}$$, there is an increasing three-term arithmetic progression of numbers from $$1$$ to $$N$$ such that either all of its terms or none of its terms are in $$S.$$

Details and assumptions

As an explicit example, if $$N = 1000$$ and $$S$$ is the subset of all primes less than 1000, then the arithmetic progression $$4-6-8$$ has none of its terms in $$S$$.

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