Editing Extremal graph theory (section)

===Homomorphism density===
{{main|Homomorphism density}}

The '''homomorphism density''' <math>t(H, G)</math> of a graph <math>H</math> in a graph <math>G</math> describes the probability that a randomly chosen map from the vertex set of <math>H</math> to the vertex set of <math>G</math> is also a [[graph homomorphism]]. It is closely related to the '''subgraph density''', which describes how often a graph <math>H</math> is found as a subgraph of <math>G</math>.

The forbidden subgraph problem can be restated as maximizing the edge density of a graph with <math>G</math>-density zero, and this naturally leads to generalization in the form of '''graph homomorphism inequalities''', which are inequalities relating <math>t(H, G)</math> for various graphs <math>H</math>.
By extending the homomorphism density to [[graphon|'''graphons''']], which are objects that arise as a limit of [[dense graph|dense graphs]], the graph homomorphism density can be written in the form of integrals, and inequalities such as the [[Cauchy-Schwarz inequality]] and [[Hölder's inequality]] can be used to derive homomorphism inequalities.

A major open problem relating homomorphism densities is [[Sidorenko's conjecture]], which states a tight lower bound on the homomorphism density of a bipartite graph in a graph <math>G</math> in terms of the edge density of <math>G</math>.