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| {{for|the category theory developed by a different author|Beck's monadicity theorem}}
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| In the context of [[discrete geometry]], '''Beck's theorem''' may refer to several different results, two of which are given below. Both appeared, alongside several other important theorems, in a well-known paper by [[József Beck]].<ref name="becks-paper"/> The two results described below primarily concern lower bounds on the number of lines ''determined'' by a set of points in the plane. (Any line containing at least two points of point set is said to be ''determined'' by that point set.)
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| ==Erdős–Beck theorem==
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| The '''Erdős–Beck theorem''' is a variation of a classical result by [[Leroy Milton Kelly|L.M. Kelly]] and W.O. J. Moser<ref>{{cite web|url=http://www.math.mcgill.ca/people/moser|title=William O. J. Moser}}</ref> involving configurations of ''n'' points of which at most ''n''−''k'' are collinear, for some 0<''k''<''O''(√''n''). They showed that if ''n'' is sufficiently large, relative to ''k'', then the configuration spans at least ''kn''−(''1/2'')(3''k''+2)(''k''−1) lines.<ref>{{citation | last1 = Kelly|first1= L. M.|author1-link=Leroy Milton Kelly|last2= Moser|first2= W. O. J. | title = On the number of ordinary lines determined by ''n'' points | journal = Canad. J. Math. | volume = 10 | year = 1958 | pages = 210–219 | url = http://www.cms.math.ca/cjm/v10/p210 | doi = 10.4153/CJM-1958-024-6}}</ref>
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| Elekes and Csaba Toth noted that the Erdős–Beck theorem does not easily extend to higher dimensions.<ref>{{cite journal|last=Elekes|first=György |author1-link=György Elekes|coauthors=Tóth, Csaba D.|author2-link=Csaba D. Tóth|year=2005|title=Incidences of not-too-degenerate hyperplanes|isbn=1-58113-991-8|journal=Proceedings of the twenty-first annual symposium on Computational geometry|publisher=Annual Symposium on Computational Geometry|location=Pisa, Italy|pages=16–21}}</ref> Take for example a set of 2''n'' points in '''R'''<sup>3</sup> all lying on two [[skew lines]]. Assume that these two lines are each incident to ''n'' points. Such a configuration of points spans only 2''n'' planes. Thus, a trivial extension to the hypothesis for point sets in '''R'''<sup>''d''</sup> is not sufficient to obtain the desired result.
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| This result was first conjectured by [[Paul Erdős|Erdős]], and proven by Beck. (See ''Theorem 5.2'' in.<ref name="becks-paper">{{cite journal|last=Beck|first=József|author-link=József Beck|year=1983|title=On the lattice property of the plane and some problems of Dirac, Motzkin, and Erdős in combinatorial geometry|journal=Combinatorica |volume=3|pages=281–297|mr=0729781|doi=10.1007/BF02579184}}</ref>)
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| ===Statement of the theorem===
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| Let ''S'' be a set of ''n'' points in the plane. If no more than ''n'' − ''k'' points lie on any line for some 0 ≤ ''k'' < ''n'' − 2, then there exist Ω(''nk'') lines determined by the points of ''S''.
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| ===Proof===
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| {{Empty section|A proof of this theorem|date=March 2010}}
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| ==Beck's theorem==
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| '''Beck's theorem''' says that finite collections of points in the plane fall into one of two extremes; one where a large fraction of points lie on a single line, and one where a large number of lines are needed to connect all the points.
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| Although not mentioned in Beck's paper, this result is implied by the [[Erdős–Beck theorem]].<ref>Beck's Theorem can be derived by letting k=n(1-1/C) and applying the Erdős–Beck theorem.</ref>
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| ===Statement of the theorem===
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| The theorem asserts the existence of positive constants ''C'', ''K'' such that that given any ''n'' points in the plane, at least one of the following statements is true: | |
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| # There is a line which contains at least ''n''/C of the points.
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| # There exist at least <math>n^2/K</math> lines, each of which contains at least two of the points.
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| In Beck's original argument, ''C'' is 100 and ''K'' is an unspecified constant; it is not known what the optimal values of ''C'' and ''K'' are.
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| === Proof ===
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| A proof of Beck's theorem can be given as follows. Consider a set ''P'' of ''n'' points in the plane. Let ''j'' be a positive integer. Let us say that a pair of points ''A'', ''B'' in the set ''P'' is ''j-connected'' if the line connecting ''A'' and ''B'' contains between <math>2^j</math> and <math>2^{j+1}-1</math> points of ''P'' (including ''A'' and ''B'').
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| From the [[Szemerédi–Trotter theorem]], the number of such lines is <math>O( n^2 / 2^{3j} + n / 2^j )</math>, as follows: Consider the set ''P'' of ''n'' points, and the set ''L'' of all those lines spanned by pairs of points of ''P'' that contain at least <math> 2^j </math> points of ''P''. Note that <math> |L| \cdot {2^j \choose 2} \leq {n \choose 2}</math>, since no two points can lie on two distinct lines. Now using [[Szemerédi–Trotter theorem]], it follows that the number of incidences between ''P'' and ''L'' is at most <math>O(n^2/2^{2j} + n)</math>. All the lines connecting ''j-connected'' points also lie in ''L'', and each contributes at least <math>2^j</math> incidences. Therefore the total number of such lines is <math>O(n^2/2^{3j} + n/2^j)</math>.
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| Since each such line connects together <math>\Omega( 2^{2j} )</math> pairs of points, we thus see that at most <math>O( n^2 / 2^j + 2^j n )</math> pairs of points can be ''j''-connected.
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| Now, let ''C'' be a large constant. By summing the [[geometric series]], we see that the number of pairs of points which are ''j''-connected for some ''j'' satisfying <math>C \leq 2^j \leq n/C</math> is at most <math>O( n^2 / C)</math>.
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| On the other hand, the total number of pairs is <math>\frac{n(n-1)}{2}</math>. Thus if we choose ''C'' to be large enough, we can find at least <math>n^2 / 4 </math> pairs (for instance) which are not ''j''-connected for any <math>C \leq 2^j \leq n/C</math>. The lines that connect these pairs either pass through fewer than ''2C'' points, or pass through more than ''n/C'' points. If the latter case holds for even one of these pairs, then we have the first conclusion of Beck's theorem. Thus we may assume that all of the <math>n^2 / 4</math> pairs are connected by lines which pass through fewer than ''2C'' points. But each such line can connect at most <math>C(2C-1)</math> pairs of points. Thus there must be at least <math>n^2 / 4C(2C-1)</math> lines connecting at least two points, and the claim follows by taking <math>K = 4C(2C-1)</math>.
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| == References ==
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| <references/>
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| {{DEFAULTSORT:Beck's Theorem (Geometry)}}
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| [[Category:Euclidean plane geometry]] | |
| [[Category:Theorems in discrete geometry]]
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| [[Category:Articles containing proofs]]
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