This question has been answered up to a logarithmic factor by Michael Payne and myself [["On the general position subset selection problem"]][1]. 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. 


  [1]: http://arxiv.org/abs/1208.5289