# 2017-08 Long arithmetic progression

Does there exist a constant $$\varepsilon>0$$ such that for each positive integer $$n$$ and each subset $$A$$ of $$\{1,2,\ldots,n\}$$ with $$\lvert A\rvert<\varepsilon n$$, there exists an artihmetic progression $$S$$ in $$\{1,2,\ldots,n\}$$ such that $$S\cap A=\emptyset$$ and $$\lvert S\rvert >\varepsilon n$$?

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