TITLE: Generalization of Sylvester-Gallai theorem
QUESTION [8 upvotes]: The Sylvester-Gallai theorem states that it is not possible to arrange a finite number
  of points so that a line through every two of them passes through a
  third unless they are all on a single line.

Is there any modern research that generalizes this theorem or finds some unexpected relations between this theorem and other parts of mathematics? Could you point out some references (preferably survey papers)?

REPLY [2 votes]: 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, 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
Forum of Mathematics, Sigma, vol. 2, 2014.
See also this blog post on fractional Gallai-Sylvester.