Here are links to some recent generalizations of the Gallai-Sylvester theorem.

1) B. Barak, Z. Dvir, A. Wigderson, A. Yehudayoff
[Fractional Sylvester-Gallai theorems,][1] Proceedings of the National Academy of Sciences of the United States of America 2012. (Link to a journal proceeding.)

2)  A. Ai, Z. Dvir, S. Saraf, A. Wigderson
[Sylvester-Gallai Type Theorems for Approximate Collinearity][2]
Forum of Mathematics, Sigma, vol. 2, 2014.


See also this blog post on [fractional Gallai-Sylvester][3].


  [1]: http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/ZEEV/fractionalsylvester-gallai.pdf
  [2]: http://www.math.ias.edu/~avi/PUBLICATIONS/AiDvSaWi2014.pdf
  [3]: https://gilkalai.wordpress.com/2012/01/20/fractional-sylvester-gallai/