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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Szemerédi–Trotter theorem ： ウィキペディア英語版 Szemerédi–Trotter theorem The Szemerédi–Trotter theorem is a mathematical result in the field of combinatorial geometry. It asserts that given points and 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 :$O\; \backslash left\; (\; n^\}\; m^\}\; +\; n\; +\; m\; \backslash right\; ),$ this bound cannot be improved, except in terms of the implicit constants. 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 :$O\; \backslash left(\; \backslash frac\; +\; \backslash frac\; \backslash right\; ).$ The original proof of Szemerédi and Trotter 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. (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 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 of the points. In particular it will contain at least ''k''/2 such line segments, since we have assumed . Adding this up over all of the 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 be the number of such line segments, it will suffice to show that :$e\; =\; O\; \backslash left\; (\; n^\}\; m^\}\; +\; n\; +\; m\; \backslash right).$ Now consider the graph formed by using the points as vertices, and the ''e'' line segments as edges. Since all of the line segments lie on one of lines, and any two lines intersect in at most one point, the crossing number of this graph is at most . Applying the crossing number inequality we thus conclude that either , or that . In either case and we obtain the desired bound :$e\; =\; O\; \backslash left\; (\; n^\}\; m^\}\; +\; n\; +\; m\; \backslash right\; ).$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Szemerédi–Trotter theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.