LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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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.
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Every finite graph $H$ with $χ (H) = r$ is an ordinary subgraph of $K_{r} [s]$ for some $s$

Statement

If a finite graph $H$ has $χ (H) = r$ , then $H$ is an ordinary subgraph of $K_{r} [s]$ for some $s \geq 1$ . For the null graph, $r = 0$ and the assertion uses the convention $K_{0} [1]$ is null.

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]

The balanced blowup $H [s]$ replaces each vertex by an independent $s$ -set and each edge by all cross edges between the corresponding parts (Ordinary-subgraph extremal number $ex (n, H)$ , Turán graph $T_{n, r}$ , and balanced blowup $H [s]$ ).

[F3]

Every finite set has a unique natural-number cardinality (The cardinality $∣ A ∣$ of a finite set).

Proof

technique · place colour classes into blowup parts

1.1

If $H$ is null, it embeds in $K_{0} [1]$ . Otherwise choose a proper colouring with colours $1, \dots, r$ and let $s \geq 1$ be the largest colour-class size. Inject each colour class into the corresponding size- $s$ independent part of $K_{r} [s]$ .

givenF1F2F3

2.1

Every edge of $H$ joins vertices of different colours, and all cross-part edges occur in $K_{r} [s]$ . The combined injection therefore preserves every edge and is an ordinary-subgraph embedding.

step 1.1givenF1F2∎

Depends on

Used by

Erdős–Stone–Simonovits: ex(n,H)=(1-1/(χ(H)-1)+o(1))binom n2 for every graph with an edge Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 16 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)

Every finite graph HH with χ(H)=rχ(H)=r is an ordinary subgraph of Kr[s]Kr​[s] for some ss