TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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Every finite graph with an edge has a Turán density $π (H) = \lim_{n \to \infty} ex (n, H) / (\binom{n}{2})$

Statement

For every finite graph $H$ with at least one edge, the limit

$π (H) : = \lim_{n \to \infty} \frac{ex (n, H)}{(\binom{n}{2})}$

exists in $[0, 1]$ . It equals

$\inf_{n \geq 2} \frac{ex (n, H)}{(\binom{n}{2})} .$

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

For a finite graph $H$ with an edge and every $n \geq 3$ , the normalized extremal numbers satisfy $ex (n, H) / (\binom{n}{2}) \leq ex (n - 1, H) / (\binom{n - 1}{2})$ ( $ex (n, H) / (\binom{n}{2})$ is nonincreasing for $n \geq 2$ ).

[F2]

Every nonincreasing real sequence bounded below converges to the infimum of its range (A nondecreasing sequence bounded above converges to the supremum of its range, and a nonincreasing sequence bounded below to the infimum).

Proof

technique · apply bounded monotone convergence

1.1

The normalized extremal numbers are nonincreasing. They lie in $[0, 1]$ because an edge count is nonnegative and no simple $n$ -vertex graph has more than $(\binom{n}{2})$ edges.

givenF1

2.1

Bounded monotone convergence makes the sequence converge to its infimum. The bounds in step 1.1 place that value in $[0, 1]$ .

step 1.1givenF2∎

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yufei Zhao, Graph Theory and Additive Combinatorics (standard reference, not scraped)

Every finite graph with an edge has a Turán density π(H)=lim⁡n→∞ex⁡(n,H)/(n2)π(H)=limn→∞​ex(n,H)/(2n​)