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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)?

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3 Answers 3

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Yes! See the beautiful recent paper of Ben Green and Terry Tao, which shows that for large $n$, any collection of $n$ points not all collinear will have at least $n/2$ ordinary lines.

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  • $\begingroup$ Could you add a comment specifying what is the required background to fully understand the paper? $\endgroup$
    – user60665
    Commented Dec 23, 2014 at 0:17
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    $\begingroup$ @Dal: The proof uses some considerations using the Euler characteristic, results on points on (cubic) curves, and ideas from additive combinatorics to force structure in configurations having few ordinary lines. You could also look at Tao's blog post: terrytao.wordpress.com/2012/08/24/… $\endgroup$
    – Lucia
    Commented Dec 23, 2014 at 0:21
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Here is a generalization to arbitrary finite metric spaces. Recall that the Sylvester-Gallai theorem easily implies the following theorem.

Theorem to be generalized. Every non-collinear set of $n$ points in the plane determines at least $n$ lines.

Note that there is a definition of line in a metric space $(X, d)$ using the notion of betweenness, which I will now describe. We say that a point $b$ is between points $a$ and $c$ if $d(a,b)+d(b,c)=d(a,c)$. The line determined by two points $a$ and $b$ is then the set of points $c$ such that $c$ is between $a$ and $b$, or $a$ is between $c$ and $b$, or $b$ is between $a$ and $c$.

Chen-Chvatál Conjecture. Every finite metric space on $n \geq 2$ points either has at least $n$ distinct lines or a universal line.

This conjecture is still wide open, although there is a sort of industry of results proving it for restricted classes of metrics. For example, there is this paper of Aboulker and Kapadia which proves the Chen-Chvatál Conjecture for metrics coming from distance-hereditary graphs.

Interestingly, it turns out that the Sylvester-Gallai theorem does not hold for all finite metric spaces. However, there are no finite metric spaces for which it is known that the Chen-Chvatál Conjecture is false.

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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.

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