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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