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	~~The '''Szemerédi–Trotter theorem''' is a [[mathematics\|mathematical]] result in the field of [[combinatorial geometry]]. It asserts that given ''n'' points and ''m'' lines in the plane~~, ~~the number of [[Incidence (geometry)\|incidences]] (i.e. the number of point-line pairs, such that the point lies on the line)~~ is		Nice to satisfy you, my name is Araceli Oquendo but I don't like when individuals use my complete name. Interviewing is what I do in my working day job. For many years she's been residing in Kansas. What he truly enjoys doing is to perform handball but he is having difficulties to discover time for it.<br><br>Feel free to surf to my web page; [http://jalic.de/index.php?mod=users&action=view&id=99 auto warranty]

	~~:<math>O( n^{2/3} m^{2/3} + n + m )</math>, which is a bound that cannot be improved, except in terms of the implicit constants.~~

	~~An equivalent formulation of the theorem is the following. Given~~ '~~'n'' points and an integer ''k'' > 2, the number of lines~~
	~~which pass through at least ''k'' of the points is~~

	~~:<math>O( n^2 / k^3 + n/k).</math>~~

	~~The original proof of [[Endre Szemerédi\|Szemerédi]] and [[William T. Trotter\|Trotter]]<ref~~ name~~="Szemerédi">{{cite journal\| last=Szemerédi \| first=Endre \| authorlink=Endre Szemerédi \| coauthors=William T~~. ~~Trotter \| year=1983 \| title=Extremal problems~~ in ~~discrete geometry \| journal=Combinatorica \| volume=3 \| doi=10~~.1007/BF02579194 \| pages=381–392 \| issue=3–4}}</ref> was somewhat complicated, using a combinatorial technique known as ''[[cell decomposition]]''. Later, Székely discovered a much simpler proof using the [[Crossing number (graph_theory)#The crossing number inequality\|crossing number inequality]] for [[Graph (mathematics)\|graphs]].<ref name="Székely">{{cite journal\| last=Székely \| first=László A. \| year=1997 \| title=Crossing numbers and hard Erdős problems in discrete geometry \| journal=Combinatorics, Probability and Computing \| volume=6 \| issue=3 \| pages=353–358 \| url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.125.1484 \| doi=10.1017/S0963548397002976}}</ref> (See below.)

	~~The Szemerédi–Trotter theorem has a number of consequences, including [[Beck~~'s ~~theorem (geometry)\|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 2''m'' incidences~~ to
	~~the total number. Thus we may assume that every line contains at least three of the points.~~

	~~If a line contains ''k'' points, then it will contain ''k''−1 line segments which connect two of~~
	~~the ''n'' points. In particular it will contain at least ''k''/2 such line segments, since we have assumed ''k''≥ 3.~~
	~~Adding this up over all of the ''m'' 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 ''e'' be the number of such line segments, it will suffice~~ to
	~~show that <math>e = O( n^{2/3} m^{2/3} + n + m )</math>.~~

	~~Now consider the [[Graph (mathematics)\|graph]] formed by using the ''n'' points as vertices, and the ''e'' line segments~~
	as edges. Since all of the line segments lie on one of ''m'' lines, and any two lines intersect in at most one point, the [[Crossing number (graph theory)\|crossing number]] of this graph is at most <math>m^2</math>. Applying the [[Crossing number (graph_theory)#The crossing number inequality\|crossing number inequality]]
	we thus conclude that either ''e'' ≤ 7.5''n'', or that ''m''<sup>2</sup> ≥ ''e''<sup>3</sup> / 33.75''n''<sup>2</sup>. In either case ''e'' ≤ 3.24''n''<sup>2 / 3</sup>''m''<sup>2 / 3</sup> + 7.5''n'' and
	~~we obtain the desired bound <math>e = O( n^{2/3} m^{2/3} + n + m )</math>.~~

	~~== Proof of the second formulation ==~~
	Since every pair of points can be connected by at most one line, there can be at most ''n''(''n'' − 1)/2 lines which can connect at ''k'' or more points, since ''k'' ≥ 2. This bound will prove the theorem when ''k'' is small
	~~(e.g. if ''k'' ≤ ''C''~~ for ~~some absolute constant ''C''). Thus, we need only consider the case when ''k'' is large, say~~
	~~''k'' ≥ ''C''~~.

	Suppose that there are ''m'' lines that each contain at least ''k'' points. These lines generate at least ''mk'' incidences, and so by the first formulation of the Szemerédi–Trotter theorem, we have
	~~:<math> mk = O( n^{2/3} m^{2/3} + n + m )</math>~~
	~~and so at least one of the statements <math>mk = O( n^{2/3} m^{2/3} )</math>, <math> mk = O(n)</math>, or~~
	<~~math~~>~~mk = O(m)~~<~~/math~~> ~~is true. The third possibility is ruled out since ''k'' was assumed~~ to ~~be large, so we are left~~
	~~with the first two. But in either of these two cases, some elementary algebra will give the bound <math>m = O( n^2 / k^3 + n/k )</math> as desired.~~

	~~== Optimality ==~~
	~~Except for its constant, the Szemerédi–Trotter incidence bound cannot be improved. To see this, consider for any positive integer <math>N\in\mathbb{Z}^+</math> a set of points on~~
	the integer [[Lattice (group)\|lattice]] <math>P = \{ (a, b) \in \mathbb{Z}^2 : 1 \leq a \leq N; 1 \leq b \leq 2N^2 \}</math> and a set of lines <math>L = \{ (x, mx + b) : m, b \in \mathbb{Z}; 1 \leq m \leq N; 1 \leq b \leq N^2\}.</math> Clearly, <math>\|P\| = 2N^3</math> and <math>\|L\| = N^3</math>. Since each line is incident to ~~<math>N</math> points (i.e., once for each <math>x \in \{1, 2, \ldots, N\}</math>), the number of incidences is <math>N^4</math> which matches the upper bound.<ref>{{cite~~ web~~\|url=~~http://~~terrytao.wordpress.com/tag/szemeredi-trotter-theorem/\|author=Terence Tao\|title=An incidence theorem in higher dimensions\|author-link=Terence Tao\|date=March 17, 2011\|accessdate=August 26, 2012}}</ref>~~

	~~==Generalization to ℝ<sup>''d''</sup>==~~
	One generalization of this result to arbitrary dimension, <big>ℝ</big><sup>''d''</sup>, was found by Agarwal and Aronov.<ref>{{cite journal\|last1=Agarwal\|first1=Pankaj\|last2=Aronov\|first2=Boris\|author1-link=Pankaj K. Agarwal\|author2-link=Boris Aronov\|title=Counting facets and incidences\|year=1992\|journal=[[Discrete and Computational Geometry]]\|publisher=Springer\|volume=7\|issue=1\|pages=359–369\|doi=10.1007/BF02187848}}</ref>
	Given a set of <math>n</math> points, <math>S</math>, and the set of <math>m</math> hyperplanes, <math>H</math>, which are each spanned by <math>S</math>, the number of incidences between <math>S</math> and <math>H</math> is bounded above by
	~~:<math>O(m^{2/3}n^{d/3}+n^{d-1})~~.</~~math>~~

	~~Equivalently, the number of hyperplanes in <math>H</math> containing <math>k</math> or more points is bounded above by~~
	~~:<math>O(n^d/k^3+n^{d-1}/k)~~.~~</math>~~

	~~A construction due to Edelsbrunner shows this bound to be asymptotically optimal.<ref>{{cite book\|title~~=~~Algorithms in Combinatorial Geometry\|last~~=~~Edelsbrunner\|first~~=~~Herbert\|author-link=Herbert Edelsbrunner\|publisher=Springer-Verlag\|year=1987\|chapter=6.5 Lower bounds for many cells\|isbn=3-540-13722-X}}</ref>~~

	~~[[József Solymosi\|Solymosi]] and [[Terence Tao\|Tao]] obtained near sharp upper bounds~~
	for the number of incidences between points and algebraic varieties in higher dimensions. Their proof uses the [[Polynomial Ham Sandwich Theorem]].<ref>{{Cite journal\|last1=Solymosi\|first1=J.\|last2=Tao\|first2=T.\|author2-link=Terence Tao\|title=An incidence theorem in higher dimensions\|date=September 2012\|journal=[[Discrete and Computational Geometry]]\|issue=2\|volume=48\|doi=10.1007/s00454-012-9420-x}}</ref>

	~~== References ==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Szemeredi-Trotter theorem}}~~
	~~[[Category:Euclidean plane geometry]]~~
	~~[[Category:Theorems in discrete geometry]]~~
	~~[[Category:Theorems in combinatorics]]~~
	~~[[Category:Articles containing proofs]~~]

en>Yobot: WP:CHECKWIKI error fixes + general fixes using AWB (7467)

2010-12-14T23:17:15Z

WP:CHECKWIKI error fixes + general fixes using AWB (7467)

New page

The '''Szemerédi–Trotter theorem''' is a [[mathematics|mathematical]] result in the field of [[combinatorial geometry]]. It asserts that given ''n'' points and ''m'' lines in the plane, the number of [[Incidence (geometry)|incidences]] (i.e. the number of point-line pairs, such that the point lies on the line) is

:<math>O( n^{2/3} m^{2/3} + n + m )</math>, which is a bound that cannot be improved, except in terms of the implicit constants.

An equivalent formulation of the theorem is the following. Given ''n'' points and an integer ''k'' > 2, the number of lines
which pass through at least ''k'' of the points is

:<math>O( n^2 / k^3 + n/k).</math>

The original proof of [[Endre Szemerédi|Szemerédi]] and [[William T. Trotter|Trotter]]<ref name="Szemerédi">{{cite journal| last=Szemerédi | first=Endre | authorlink=Endre Szemerédi | coauthors=William T. Trotter | year=1983 | title=Extremal problems in discrete geometry | journal=Combinatorica | volume=3 | doi=10.1007/BF02579194 | pages=381–392 | issue=3–4}}</ref> was somewhat complicated, using a combinatorial technique known as ''[[cell decomposition]]''. Later, Székely discovered a much simpler proof using the [[Crossing number (graph_theory)#The crossing number inequality|crossing number inequality]] for [[Graph (mathematics)|graphs]].<ref name="Székely">{{cite journal| last=Székely | first=László A. | year=1997 | title=Crossing numbers and hard Erdős problems in discrete geometry | journal=Combinatorics, Probability and Computing | volume=6 | issue=3 | pages=353–358 | url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.125.1484 | doi=10.1017/S0963548397002976}}</ref> (See below.)

The Szemerédi–Trotter theorem has a number of consequences, including [[Beck's theorem (geometry)|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 2''m'' incidences to
the total number. Thus we may assume that every line contains at least three of the points.

If a line contains ''k'' points, then it will contain ''k''−1 line segments which connect two of
the ''n'' points. In particular it will contain at least ''k''/2 such line segments, since we have assumed ''k''≥ 3.
Adding this up over all of the ''m'' 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 ''e'' be the number of such line segments, it will suffice to
show that <math>e = O( n^{2/3} m^{2/3} + n + m )</math>.

Now consider the [[Graph (mathematics)|graph]] formed by using the ''n'' points as vertices, and the ''e'' line segments
as edges. Since all of the line segments lie on one of ''m'' lines, and any two lines intersect in at most one point, the [[Crossing number (graph theory)|crossing number]] of this graph is at most <math>m^2</math>. Applying the [[Crossing number (graph_theory)#The crossing number inequality|crossing number inequality]]
we thus conclude that either ''e'' ≤ 7.5''n'', or that ''m''2 ≥ ''e''3 / 33.75''n''2. In either case ''e'' ≤ 3.24''n''2 / 3''m''2 / 3 + 7.5''n'' and
we obtain the desired bound <math>e = O( n^{2/3} m^{2/3} + n + m )</math>.

== Proof of the second formulation ==
Since every pair of points can be connected by at most one line, there can be at most ''n''(''n'' − 1)/2 lines which can connect at ''k'' or more points, since ''k'' ≥ 2. This bound will prove the theorem when ''k'' is small
(e.g. if ''k'' ≤ ''C'' for some absolute constant ''C''). Thus, we need only consider the case when ''k'' is large, say
''k'' ≥ ''C''.

Suppose that there are ''m'' lines that each contain at least ''k'' points. These lines generate at least ''mk'' incidences, and so by the first formulation of the Szemerédi–Trotter theorem, we have
:<math> mk = O( n^{2/3} m^{2/3} + n + m )</math>
and so at least one of the statements <math>mk = O( n^{2/3} m^{2/3} )</math>, <math> mk = O(n)</math>, or
<math>mk = O(m)</math> is true. The third possibility is ruled out since ''k'' was assumed to be large, so we are left
with the first two. But in either of these two cases, some elementary algebra will give the bound <math>m = O( n^2 / k^3 + n/k )</math> as desired.

== Optimality ==
Except for its constant, the Szemerédi–Trotter incidence bound cannot be improved. To see this, consider for any positive integer <math>N\in\mathbb{Z}^+</math> a set of points on
the integer [[Lattice (group)|lattice]] <math>P = \{ (a, b) \in \mathbb{Z}^2 : 1 \leq a \leq N; 1 \leq b \leq 2N^2 \}</math> and a set of lines <math>L = \{ (x, mx + b) : m, b \in \mathbb{Z}; 1 \leq m \leq N; 1 \leq b \leq N^2\}.</math> Clearly, <math>|P| = 2N^3</math> and <math>|L| = N^3</math>. Since each line is incident to <math>N</math> points (i.e., once for each <math>x \in \{1, 2, \ldots, N\}</math>), the number of incidences is <math>N^4</math> which matches the upper bound.<ref>{{cite web|url=http://terrytao.wordpress.com/tag/szemeredi-trotter-theorem/|author=Terence Tao|title=An incidence theorem in higher dimensions|author-link=Terence Tao|date=March 17, 2011|accessdate=August 26, 2012}}</ref>

==Generalization to ℝ''d''==
One generalization of this result to arbitrary dimension, <big>ℝ</big>''d'', was found by Agarwal and Aronov.<ref>{{cite journal|last1=Agarwal|first1=Pankaj|last2=Aronov|first2=Boris|author1-link=Pankaj K. Agarwal|author2-link=Boris Aronov|title=Counting facets and incidences|year=1992|journal=[[Discrete and Computational Geometry]]|publisher=Springer|volume=7|issue=1|pages=359–369|doi=10.1007/BF02187848}}</ref>
Given a set of <math>n</math> points, <math>S</math>, and the set of <math>m</math> hyperplanes, <math>H</math>, which are each spanned by <math>S</math>, the number of incidences between <math>S</math> and <math>H</math> is bounded above by
:<math>O(m^{2/3}n^{d/3}+n^{d-1}).</math>

Equivalently, the number of hyperplanes in <math>H</math> containing <math>k</math> or more points is bounded above by
:<math>O(n^d/k^3+n^{d-1}/k).</math>

A construction due to Edelsbrunner shows this bound to be asymptotically optimal.<ref>{{cite book|title=Algorithms in Combinatorial Geometry|last=Edelsbrunner|first=Herbert|author-link=Herbert Edelsbrunner|publisher=Springer-Verlag|year=1987|chapter=6.5 Lower bounds for many cells|isbn=3-540-13722-X}}</ref>

[[József Solymosi|Solymosi]] and [[Terence Tao|Tao]] obtained near sharp upper bounds
for the number of incidences between points and algebraic varieties in higher dimensions. Their proof uses the [[Polynomial Ham Sandwich Theorem]].<ref>{{Cite journal|last1=Solymosi|first1=J.|last2=Tao|first2=T.|author2-link=Terence Tao|title=An incidence theorem in higher dimensions|date=September 2012|journal=[[Discrete and Computational Geometry]]|issue=2|volume=48|doi=10.1007/s00454-012-9420-x}}</ref>

== References ==
{{reflist}}

{{DEFAULTSORT:Szemeredi-Trotter theorem}}
[[Category:Euclidean plane geometry]]
[[Category:Theorems in discrete geometry]]
[[Category:Theorems in combinatorics]]
[[Category:Articles containing proofs]]