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This question has been answered up to a logarithmic factor by Michael Payne and myself ["On the general position subset selection problem"]. We show that $n \leqslant c m^2 \log m$. The proof employs the Szemeredi-Trotter Theorem to bound the number of collinear triples in a point set. Then we apply known results about independent sets in 3-uniform hypergraphs to conclude the result.