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Beck's theorem
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.〔 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 L.M. Kelly and W.O. J. Moser 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.
Elekes and Csaba Toth noted that the Erdős–Beck theorem does not easily extend to higher dimensions. Take for example a set of 2''n'' points in R3 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''d'' is not sufficient to obtain the desired result.
This result was first conjectured by Erdős, and proven by Beck. (See ''Theorem 5.2'' in.)
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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