DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Edge density and the asymptotic notations $O$ , $o$ , $Ω$ , and $Θ$ for extremal functions

Definition

For an $n$ -vertex graph $G$ with $n \geq 2$ , its edge density is

$d (G) : = \frac{e (G)}{(\binom{n}{2})} .$

The normalized extremal number is $ex (n, H) / (\binom{n}{2})$ for $n \geq 2$ .

For eventually nonnegative functions $f, g : N \to R$ with $g (n) > 0$ eventually:

$f = O (g)$ means that some $C, N > 0$ satisfy $f (n) \leq C g (n)$ for $n \geq N$ ;
$f = o (g)$ means $f (n) / g (n) \to 0$ ;
$f = Ω (g)$ means that some $c, N > 0$ satisfy $f (n) \geq c g (n)$ for $n \geq N$ ;
$f = Θ (g)$ means both $f = O (g)$ and $f = Ω (g)$ .

A subscript, as in $O_{r, s}$ , permits the hidden constant and threshold to depend on the subscripted parameters. No normalized edge density is assigned when $n < 2$ because $(\binom{n}{2}) = 0$ .

Depends on

Used by

Dependency tree · two levels

15 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)

Edge density and the asymptotic notations OO, oo, ΩΩ, and ΘΘ for extremal functions