Editing Extremal graph theory (section)

[[File:Turan 13-4.svg|thumb|The [[Turán graph]] ''T''(''n'',''r'') is an example of an extremal graph. It has the maximum possible number of edges for a graph on ''n'' vertices without (''r''&nbsp;+&nbsp;1)-[[clique (graph theory)|cliques]]. This is ''T''(13,4).]]

'''Extremal graph theory''' is a branch of [[combinatorics]], itself an area of [[mathematics]], that lies at the intersection of [[extremal combinatorics]] and [[graph theory]]. In essence, extremal graph theory studies how global properties of a graph influence local substructure.
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Results in extremal graph theory deal with quantitative connections between various [[Graph property|graph properties]], both global (such as the number of vertices and edges) and local (such as the existence of specific subgraphs), and problems in extremal graph theory can often be formulated as optimization problems: how big or small a parameter of a graph can be, given some constraints that the graph has to satisfy?
<ref name="pcm" >
{{Princeton Companion to Mathematics 
  |article=Extremal and Probabilistic Combinatorics 
  |pages=562-575 
  |author1-surname=Alon 
  |author1-given=Noga
  |author1-link=Noga Alon
  |author2-surname=Krivelevich
  |author2-given=Michael
  |author2-link=Michael Krivelevich
}}
</ref>
A graph that is an optimal solution to such an optimization problem is called an '''extremal graph''', and extremal graphs are important objects of study in extremal graph theory.

Extremal graph theory is closely related to fields such as [[Ramsey theory]], [[spectral graph theory]], [[computational complexity theory]], and [[additive combinatorics]], and frequently employs the [[probabilistic method]].