In
discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic ge ...
, Beck's theorem is any of several different results, two of which are given below. Both appeared, alongside several other important theorems, in a well-known paper by
József Beck
József Beck (Budapest, Hungary, February 14, 1952) is a Harold H. Martin Professor of Mathematics at Rutgers University.
His contributions to combinatorics include the partial colouring lemma and the Beck–Fiala theorem in '' discrepancy th ...
.
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''(). 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 R
3 all lying on two
skew lines
In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the sa ...
. 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.
)
Statement
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
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.
[Beck's Theorem can be derived by letting ''k'' = ''n''(1 − 1/''C'') and applying the Erdős–Beck theorem.]
Statement
The theorem asserts the existence of positive constants ''C'', ''K'' such 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
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
and
points of ''P'' (including ''A'' and ''B'').
From the
Szemerédi–Trotter theorem, the number of such lines is
, 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
points of ''P''. 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
. All the lines connecting ''j-connected'' points also lie in ''L'', and each contributes at least
incidences. Therefore, the total number of such lines is
.
Since each such line connects together
pairs of points, we thus see that at most
pairs of points can be ''j''-connected.
Now, let ''C'' be a large constant. By summing the
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each suc ...
, we see that the number of pairs of points which are ''j''-connected for some ''j'' satisfying
is at most
.
On the other hand, the total number of pairs is
. Thus if we choose ''C'' to be large enough, we can find at least
pairs (for instance) which are not ''j''-connected for any
. The lines that connect these pairs either pass through fewer than 2''C'' 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
pairs are connected by lines which pass through fewer than 2''C'' points. But each such line can connect at most
pairs of points. Thus there must be at least
lines connecting at least two points, and the claim follows by taking
.
References
{{DEFAULTSORT:Beck's Theorem (Geometry)
Euclidean plane geometry
Theorems in discrete geometry
Articles containing proofs