ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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$K_{3, 4}$ realises $ex (7, K_{3}) = 12$

Example

The complete bipartite graph $K_{3, 4}$ is triangle-free and has $3 \cdot 4 = 12$ edges. Hence it realizes

$ex (7, K_{3}) = ⌊ \frac{49}{4} ⌋ = 12.$

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

The complete bipartite graph $K_{A, B}$ has exactly all edges joining a vertex of $A$ to a vertex of $B$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

For every $n \in N$ , Mantel's theorem gives $ex (n, K_{3}) = ⌊ n^{2} / 4 ⌋$ , and a triangle-free $n$ -vertex graph attains equality exactly when it is the balanced complete bipartite graph up to isomorphism (Mantel's theorem: $ex (n, K_{3}) = ⌊ n^{2} / 4 ⌋$ , uniquely attained by $T_{n, 2}$ ).

Verification

technique · count cross edges and apply Mantel

1.1

Every edge of $K_{3, 4}$ crosses its bipartition, so a three-vertex cycle is impossible, and there are exactly $3 \cdot 4 = 12$ possible cross edges.

givenF1

2.1

Mantel's theorem gives the matching upper bound $⌊ 7^{2} / 4 ⌋ = 12$ , so the graph is extremal.

step 1.1givenF2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 8 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)

K3,4K3,4​ realises ex⁡(7,K3)=12ex(7,K3​)=12