Addressing the simplest case of ordinary lines in Sylvester Gallai theorem

Let P be a set of n points on the plane \mathbb{R}^2. We call a line on this plane k-rich if it passes through exactly k many points of P.

Usually 2-rich lines are also called ordinary lines.

Then the simplest version of Sylvester-Gallai theorem says that

Given any finite \ge 2 set of points on the plane such that not all are collinear, then there must be a 2-rich line.

It was proposed as an exercise problem by Sylvester in 1893 and first proved by Gallai in 1944.

The following sketch is popularly knows as Kelly’s proof of the Sylvester Gallai problem (named after Leroy Milton Kelly).

We would not mention a proof in words (as doing would only make the proof look mundane and not help you appreciate the beauty of the proof). Instead we would mention a wordless proof by providing a simple drawing that speaks the proof itself.

(Look for the shortest distance between a point in P and a line that passes through two or more points in P).

Related image

(picture taken from AMS blog).

According to prof. Terence Tao : this proof is definitely beautiful but is too clever to extend it to prove stronger and / or similar results.

Over other fields

But this result is not true in general if the reals are replaced by other fields. A trivial counterexample are the p-adic fields (p \ge 3) \mathbb{F}_p and we can take P= \mathbb{F}_{p} the entire set. A less trivial counterexample can be created using the the complex field as mentioned by prof. Terence Tao (in a lecture on topics related to incidence combinatorics).

So interestingly enough this shows some salient feature of the real numbers (although it remains disguised in the above mentioned proof), and this motivates a study of the problem more closely.

Other similar results are related to the mathematical topic “incidence combinatorics” and “algebraic combinatorics”.