Szemerédi–Trotter theorem

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In mathematics, the Szemerédi–Trotter theorem is a result in the field of combinatorial geometry. It asserts that given n points and m lines in the plane, the number of incidences (i.e. the number of point-line pairs, such that the point lies on the line) is

O(n^{2 / 3}m^{2 / 3} + n + m), which is a bound that cannot be improved, except in terms of the implicit constants.

An equivalent formulation of the theorem is the following. Given n points and an integer k > 2, the number of lines which pass through at least k of the points is

The original proof of Szemerédi and Trotter^[1] was somewhat complicated, using a combinatorial technique known as cell decomposition. Later, Szekély discovered a much simpler proof using crossing numbers of graphs.^[2] (See below.)

The Szemerédi–Trotter theorem has a number of consequences, including Beck's theorem in incidence geometry.

[edit] Proof of the first formulation

We may discard the lines which contain two or fewer of the points, as they can contribute at most 2m incidences to the total number. Thus we may assume that every line contains at least three of the points.

If a line contains k points, then it will contain k-1 line segments which connect two of the n points. In particular it will contain at least k/2 such line segments, since we have assumed k≥ 3. Adding this up over all of the m lines, we see that the number of line segments obtained in this manner is at least half of the total number of incidences. Thus if we let e be the number of such line segments, it will suffice to show that e = O(n^{2 / 3}m^{2 / 3} + n + m).

Now consider the graph formed by using the n points as vertices, and the e line segments as edges. Since all of the line segments lie on one of m lines, and any two lines intersect in at most point, the crossing number of this graph is at most m². Applying the crossing number inequality we thus conclude that either , or that . In either case we obtain the desired bound e = O(n^{2 / 3}m^{2 / 3} + n + m).

[edit] Proof of the second formulation

Since every pair of points can be connected by at most one line, there can be at most lines which can connect at k or more points, since k ≥ 2. This bound will prove the theorem when k is small (e.g. if k ≤ C for some absolute constant C). Thus, we need only consider the case when k is large, say k ≥ C.

Suppose that there are m lines that each contain at least k points. These lines generate at least mk incidences, and so by the first formulation of the Szemerédi–Trotter theorem, we have

mk = O(n^{2 / 3}m^{2 / 3} + n + m)

and so at least one of the statements mk = O(n^{2 / 3}m^{2 / 3}), mk = O(n), or mk = O(m) is true. The third possibility is ruled out since k was assumed to be large, so we are left with the first two. But in either of these two cases, some elementary algebra will give the bound m = O(n² / k³ + n / k) as desired.

[edit] References