The Szemerédi–Trotter theorem is a mathematical result in the field of Discrete geometry. It asserts that given points and lines in the Euclidean plane, the number of incidences (i.e., the number of point-line pairs, such that the point lies on the line) is This bound cannot be improved, except in terms of the implicit constants. As for the implicit constants, it was shown by János Pach, Radoš Radoičić, Gábor Tardos, and Géza Tóth that the upper bound holds. Since then better constants are known due to better crossing lemma constants; the current best is 2.44. On the other hand, Pach and Tóth showed that the statement does not hold true if one replaces the coefficient 2.5 with 0.42. An equivalent formulation of the theorem is the following. Given points and an integer , the number of lines which pass through at least of the points is The original proof of Endre Szemerédi and William T. Trotter was somewhat complicated, using a combinatorial technique known as cell decomposition. Later, László Székely discovered a much simpler proof using the crossing number inequality for graphs. (See below.) The Szemerédi–Trotter theorem has a number of consequences, including Beck's theorem in incidence geometry and the Erdős-Szemerédi sum-product problem in additive combinatorics. Proof of the first formulation We may discard the lines which contain two or fewer of the points, as they can contribute at most incidences to the total number. Thus we may assume that every line contains at least three of the points. If a line contains points, then it will contain line segments which connect two consecutive points along the line. Because after discarding the two-point lines, it follows that , so the number of these line segments on each line is at least half the number of incidences on that line. Summing over all of the lines, the number of these line segments is again at least half the total number of incidences. Thus if denotes the number of such line segments, it will suffice to show that Now consider the graph formed by using the points as vertices, and the line segments as edges. Since each line segment lies on one of lines, and any two lines intersect in at most one point, the crossing number of this graph is at most the number of points where two lines intersect, which is at most . The crossing number inequality implies that