Combinatorics Extremal combinatorics Theorem

Turán's theorem

Statement

Let $n \ge r \ge 1$ be integers, and let $G$ be a simple graph with $n$ vertices that has no clique with $r + 1$ vertices. Then $G$ has at most as many edges as the Turán graph $T(n, r)$ . In particular, $G$ has at most $\left(1 - \frac{1}{r}\right) \frac{n^2}{2}$ edges.

Remarks

The bound is sharp, because $T(n, r)$ has no clique with $r + 1$ vertices. The case $r = 2$ is Mantel's theorem on graphs without triangles. The theorem is a central result of extremal graph theory.

History

Pál Turán described the theorem and the Turán graphs in 1941.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Clique definition
Turán graph example
Turán's theorem statement

Tag	`0JG1KZSPFZ`
URI	`m2://0JG1KZSPFZ`
Kind	statement, theorem
Writing	W1 stated
MSC	05C35
Status	This copy of the site has no build data. Run `m2 status 0JG1KZSPFZ`.
Uses	Clique, Turán graph
Same as	Wikidata Q1047749 Wikipedia Turán's theorem

Turán's theorem

Statement

Remarks

History

Reading path

Further reading

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