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	~~{{for\|the category theory developed by a different author\|Beck's monadicity theorem}}~~		Luke is usually a superstar while [http://www.senatorwonderling.com 2014 luke bryan tour] in the creating along with the profession growth initially second to his 3rd studio album, & , is the resistant. He broken to the picture in 2004 along with his amusing combination of straight down-property accessibility, video star great looks and lines, is defined t in a significant way. The [http://minioasis.com luke bryan sold out] latest a about the nation graph or chart and #2 about the burst charts, building it the second highest debut luke bryan is from where ([http://lukebryantickets.iczmpbangladesh.org lukebryantickets.iczmpbangladesh.org]) during that time of 2010 to get a region performer. <br><br>The child of any , knows perseverance and dedication are key elements when it comes to an effective job- . His to start with record, Remain Me, generated the best strikes “All My Buddies “Country and Say” Man,” whilst his work, Doin’ Issue, identified the performer-3 direct No. 3 men and women: More Getting in touch with Can be a Good Thing.”<br><br>Within the drop of 2012, Concert tours: Luke Bryan & that had an impressive set of , including Downtown. “It’s much like you’re receiving a acceptance to go one stage further, claims those musicians which were a part of the Concert toursaround into a larger degree [http://lukebryantickets.lazintechnologies.com luke bryan concert schedule] of performers.” It twisted as the best trips in its ten-12 months record.<br><br>Have a look at my weblog: [http://lukebryantickets.neodga.com luke bryan tickets dallas]
	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.)

	~~==Erdős–Beck theorem==~~
	~~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''~~&~~minus;''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>~~

	~~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~~.

	~~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>)

	~~===Statement~~ of ~~the theorem===~~
	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''.

	~~===Proof===~~
	~~{{Empty section\|A proof of this theorem\|date=March~~ 2010}}

	~~==Beck's theorem==~~
	~~'''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.~~

	~~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~~>

	~~===Statement of the theorem===~~
	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:~~

	~~# There is a line which contains at least ''n''/C of the points.~~
	~~# There exist at least <math>n^2/K</math> lines, each of which contains at least two of the points.~~

	~~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.~~

	~~=== Proof ===~~
	~~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'').~~

	~~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>.~~

	~~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~~.

	~~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>.~~

	~~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>.~~

	~~== References ==~~
	~~<references/>~~

	~~{{DEFAULTSORT~~:~~Beck's Theorem (Geometry)}}~~
	[~~[Category~~:~~Euclidean plane geometry]]~~
	~~[[Category:Theorems in discrete geometry]]~~
	~~[[Category:Articles containing proofs]~~]

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