Szemerédi–Trotter theorem

The Szemerédi–Trotter theorem is a mathematical 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

$O \left ( n^{\frac{2}{3}} m^{\frac{2}{3}} + n + m \right ),$

this bound 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

$O \left( \frac{n^2}{k^3} + \frac{n}{k} \right ).$

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

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

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

$e = O \left ( n^{\frac{2}{3}} m^{\frac{2}{3}} + n + m \right).$

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 one point, the crossing number of this graph is at most m². Applying the crossing number inequality we thus conclude that either e ≤ 7.5n, or that m² ≥ e³ / 33.75n². In either case e ≤ 3.24(nm)^2/3 + 7.5n and we obtain the desired bound

$e = O \left ( n^{\frac{2}{3}} m^{\frac{2}{3}} + n + m \right ).$

Proof of the second formulation

Since every pair of points can be connected by at most one line, there can be at most n(n − 1)/2 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

$mk = O \left ( n^{\frac{2}{3}} m^{\frac{2}{3}} + n + m \right ),$

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 $m = O( n^2 / k^3 + n/k )$ as desired.

Optimality

Except for its constant, the Szemerédi–Trotter incidence bound cannot be improved. To see this, consider for any positive integer N ∈ Z⁺ a set of points on the integer lattice

$P = \left \{ (a, b) \in \mathbf{Z}^2 \ : \ 1 \leq a \leq N; 1 \leq b \leq 2N^2 \right \},$

and a set of lines

$L = \left \{ (x, mx + b) \ : \ m, b \in \mathbf{Z}; 1 \leq m \leq N; 1 \leq b \leq N^2 \right \}.$

Clearly, $|P| = 2N^3$ and $|L| = N^3$. Since each line is incident to N points (i.e., once for each $x \in \{1, \cdots, N\}$), the number of incidences is $N^4$ which matches the upper bound.^[3]

Generalization to R^d

One generalization of this result to arbitrary dimension, R^d, was found by Agarwal and Aronov.^[4] Given a set of n points, S, and the set of m hyperplanes, H, which are each spanned by S, the number of incidences between S and H is bounded above by

$O \left (m^{\frac{2}{3}}n^{\frac{d}{3}}+n^{d-1} \right ).$

Equivalently, the number of hyperplanes in H containing k or more points is bounded above by

$O\left( \frac{n^d}{k^3} + \frac{n^{d-1}}{k} \right ).$

A construction due to Edelsbrunner shows this bound to be asymptotically optimal.^[5]

Solymosi and Tao obtained near sharp upper bounds for the number of incidences between points and algebraic varieties in higher dimensions. Their proof uses the Polynomial Ham Sandwich Theorem.^[6]

References

1. Szemerédi, Endre; Trotter, William T. (1983). "Extremal problems in discrete geometry". Combinatorica 3 (3–4): 381–392. doi:10.1007/BF02579194. MR 0729791.
2. Székely, László A. (1997). "Crossing numbers and hard Erdős problems in discrete geometry". Combinatorics, Probability and Computing 6 (3): 353–358. doi:10.1017/S0963548397002976. MR 1464571.
3. Terence Tao (March 17, 2011). "An incidence theorem in higher dimensions". Retrieved August 26, 2012.
4. Agarwal, Pankaj; Aronov, Boris (1992). "Counting facets and incidences". Discrete and Computational Geometry (Springer) 7 (1): 359–369. doi:10.1007/BF02187848.
5. Edelsbrunner, Herbert (1987). "6.5 Lower bounds for many cells". Algorithms in Combinatorial Geometry. Springer-Verlag. ISBN 3-540-13722-X.
6. Solymosi, J.; Tao, T. (September 2012). "An incidence theorem in higher dimensions". Discrete and Computational Geometry 48 (2). doi:10.1007/s00454-012-9420-x.