TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Erdős–Stone–Simonovits: $ex (n, H) = (1 - 1 / (χ (H) - 1) + o (1)) (\binom{n}{2})$ for every graph with an edge

Statement

Let $H$ be a finite graph with at least one edge and put $r = χ (H) \geq 2$ . Then

$ex (n, H) = (1 - \frac{1}{r - 1} + o (1)) (\binom{n}{2}) .$

Equivalently,

$\lim_{n \to \infty} \frac{ex (n, H)}{(\binom{n}{2})} = 1 - \frac{1}{χ (H) - 1} .$

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

A proper $k$ -vertex-colouring is a map $c : V \to k$ with $c (u) \neq c (v)$ for every edge ${u, v}$ , its fibres are the colour classes, and $χ (G) = \min {k \in N : G is k -colourable}$ (Proper vertex colourings and chromatic number).

[F2]

For $n \in N$ and $r \geq 1$ , Turán's theorem gives $ex (n, K_{r + 1}) = e (T_{n, r})$ , and an $n$ -vertex $K_{r + 1}$ -free graph attains equality exactly when it is isomorphic to $T_{n, r}$ (Turán's theorem with equality: $ex (n, K_{r + 1}) = e (T_{n, r})$ , and $T_{n, r}$ is the unique extremal graph).

[F3]

Every finite graph of chromatic number $r$ embeds as an ordinary subgraph of $K_{r} [s]$ for some $s \geq 1$ (Every finite graph $H$ with $χ (H) = r$ is an ordinary subgraph of $K_{r} [s]$ for some $s$ ).

[F4]

For $r \geq 2$ and $s \geq 1$ , $π (K_{r} [s]) = 1 - 1 / (r - 1)$ (Erdős–Stone for balanced blowups: $π (K_{r} [s]) = 1 - 1 / (r - 1)$ for $r \geq 2$ ).

Proof

technique · sandwich $H$ between a Turán graph and a balanced blowup

1.1

Every $(r - 1)$ -partite graph is $H$ -free, since every subgraph of it is $(r - 1)$ -colourable while $χ (H) = r$ . Therefore $T_{n, r - 1}$ gives ${lim inf}_{n \to \infty} ex (n, H) / (\binom{n}{2}) \geq 1 - 1 / (r - 1)$ .

givenF1F2

1.2

The embedding lemma gives an $s \geq 1$ with $H \subseteq K_{r} [s]$ . Hence every $H$ -free graph is $K_{r} [s]$ -free, and balanced-blowup Erdős–Stone gives ${lim sup}_{n \to \infty} ex (n, H) / (\binom{n}{2}) \leq 1 - 1 / (r - 1)$ .

givenF3F4

2.1

The two bounds agree, proving the limit and the $o (1)$ formulation. When $r = 2$ , the expression is $0$ and the same proof uses $T_{n, 1}$ for the lower bound and $K_{2} [s]$ for the upper bound, so the bipartite boundary is included.

step 1.1step 1.2givenF4∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 43 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Yufei Zhao, Graph Theory and Additive Combinatorics (standard reference, not scraped)
Reinhard Diestel, Graph Theory, Chapter 7 (standard reference, not scraped)

Erdős–Stone–Simonovits: ex⁡(n,H)=(1−1/(χ(H)−1)+o(1))(n2)ex(n,H)=(1−1/(χ(H)−1)+o(1))(2n​) for every graph with an edge