Home > Sylvester-Gallai theorem

1. All the points are collinear; or
2. There is a line which contains exactly two of the points.

This theorem was posed as a problem by James Joseph Sylvester in 1893, and solved by Tibor Gallai in 1944. A more quantitative variant of this theorem is Beck's theorem. The theorem is not true if there are infinitely many points; consider for instance the lattice .

Proof of the Sylvester-Gallai theorem

We shall use the method of infinite descent. Suppose we have a finite number of points which are not collinear (in particular, we must have at least three points). Define a connecting line to be a line which contains at least two points in the collection; we then have to show that at least one connecting line contains exactly two points.

Let l be a connecting line. Since the points are not collinear, there is at least one point P which is not on l. If l contains exactly two points, we are done. Otherwise, we know that l contains at least three points, say A, B, and C. We may assume without loss of generality that B lies between A and C. Since the angles and add up to 180 degrees, they cannot both be obtuse; without loss of generality we may assume is not obtuse (i.e. either acute or right-angled).

Now let m be the line connecting A and P. Then m is a connecting line which does not contain B. Furthermore, the distance between B and m is less than the distance between P and l.

A picture is needed here.

To summarize so far, we have started with a connecting line l and a point P not on this line, and observed that either l contains exactly two points, or that there exists another connecting line m and a point B not on that line such that the distance between B and m is less than the distance between P and l. In the latter case we simply repeat the argument again, but with P and l replaced by B and m. We cannot continue indefinitely because we would obtain an infinite decreasing sequence of distances, but the number of possible distances between points and connecting lines is finite because the original set was assumed to be finite. Thus we must eventually obtain a connecting line with exactly two points only.