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

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Above Turán density, a graph contains a positive-density family of copies of the forbidden graph

Statement

Let $H$ be a finite graph with $h$ vertices and at least one edge. For every $ε > 0$ there are $δ > 0$ and $N$ such that every $n \geq N$ graph $G$ with

$e (G) \geq (π (H) + ε) (\binom{n}{2})$

contains at least $δ n^{h}$ injective ordinary-subgraph embeddings of $H$ into $G$ .

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

For every finite graph $H$ with an edge, the normalized extremal numbers converge to $π (H)$ , their infimum over $n \geq 2$ (Every finite graph with an edge has a Turán density $π (H) = \lim_{n \to \infty} ex (n, H) / (\binom{n}{2})$ ).

[F2]

For a finite incidence relation, the sum of its row-fibre sizes equals the sum of its column-fibre sizes (Double counting: $\sum_{x \in X} ∣ R_{x} ∣ = ∣ R ∣ = \sum_{y \in Y} ∣ R^{y} ∣$ for a relation between finite sets).

[F3]

$(\binom{n}{k})$ is the number of $k$ -element subsets of an $n$ -element set (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ).

[F4]

$ex (n, H)$ is the maximum edge count of an $n$ -vertex graph with no ordinary copy of $H$ (Ordinary-subgraph extremal number $ex (n, H)$ , Turán graph $T_{n, r}$ , and balanced blowup $H [s]$ ).

Proof

technique · average over fixed-size vertex subsets

1.1

If $π (H) + ε > 1$ , take $δ = 1$ and $N = 2$ : for $n \geq 2$ one has $(\binom{n}{2}) \geq 1$ , so $(π (H) + ε) (\binom{n}{2}) > (\binom{n}{2}) \geq e (G)$ and no graph satisfies the edge hypothesis. The threshold cannot be lowered to $1$ , because $(\binom{1}{2}) = 0$ makes the hypothesis vacuous at $n = 1$ while the conclusion there demands $δ n^{h} = 1$ embedding of an $h$ -vertex $H$ into a one-vertex graph. Hence assume $π (H) + ε \leq 1$ , and choose $m \geq \max (h, 2)$ with $ex (m, H) / (\binom{m}{2}) < π (H) + ε / 2$ . For an $n$ -vertex $G$ satisfying the hypothesis, the average edge density of its induced $m$ -vertex subgraphs equals $e (G) / (\binom{n}{2}) \geq π (H) + ε$ : each edge lies in $(\binom{n - 2}{m - 2})$ such subsets.

givenF1F2F3F4

2.1

Let $p$ be the fraction of $m$ -subsets inducing more than $ex (m, H)$ edges. The remaining subsets have density below $π (H) + ε / 2$ , while every density is at most $1$ . Therefore $π (H) + ε \leq (1 - p) (π (H) + ε / 2) + p$ , so $p \geq ε / 2$ after weakening the resulting positive lower bound if necessary. Each good subset induces an $m$ -vertex graph with more than $ex (m, H)$ edges, so by [F4] it is not $H$ -free: it admits an injective ordinary-subgraph embedding of $H$ .

step 1.1givenF4

3.1

Count pairs consisting of a good $m$ -set and a chosen injective copy of $H$ inside it. There are at least $(ε / 2) (\binom{n}{m})$ pairs after choosing one copy in each good set, while any fixed embedding lies in $(\binom{n - h}{m - h})$ $m$ -sets. Thus the number of embeddings is at least $(ε / 2) (\binom{n}{m}) / (\binom{n - h}{m - h}) = (ε / 2) (\binom{n}{h}) / (\binom{m}{h})$ . For $n \geq 2 h$ , this is at least $δ n^{h}$ for some $δ > 0$ depending only on $H, ε$ .

step 2.1givenF2F3

4.1

Taking $N \geq \max (m, 2 h)$ completes the assertion with the constants constructed above.

step 1.1step 3.1∎

Depends on

Used by

Erdős–Stone for balanced blowups: π(Kᵣ[s])=1-1/(r-1) for r≥2 Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 69 results over 22 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)