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

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Erdős–Stone–Simonovits determines the extremal number for every graph

False Statement

Erdős–Stone–Simonovits by itself determines $ex (n, H)$ , even up to its order of growth, for every finite graph $H$ with an edge.

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]

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).

[F3]

If $H$ is a finite graph with an edge and $r = χ (H)$ , then $ex (n, H) = (1 - 1 / (r - 1) + o (1)) (\binom{n}{2})$ (Erdős–Stone–Simonovits: $ex (n, H) = (1 - 1 / (χ (H) - 1) + o (1)) (\binom{n}{2})$ for every graph with an edge).

[F4]

For $s, t \geq 1$ , the Kővári–Sós–Turán theorem gives $ex (N, K_{s, t}) = O_{s, t} (N^{2 - 1 / s})$ (Kővári–Sós–Turán: exact bipartite and ordinary-graph upper bounds for excluding $K_{s, t}$ ).

[F5]

$f = O (g)$ means an eventual constant upper bound, $f = o (g)$ means $f / g \to 0$ , and subscripts permit the constants and thresholds to depend on those parameters (Edge density and the asymptotic notations $O$ , $o$ , $Ω$ , and $Θ$ for extremal functions).

Refutation

technique · compare the quadratic-density conclusion with a sharper local bound

1.1

The graph $K_{2, 2}$ is bipartite, so $χ (K_{2, 2}) = 2$ . Applied to it, Erdős–Stone–Simonovits says only $ex (n, K_{2, 2}) = o (n^{2})$ .

givenF1F2F3F5

1.2

The separate common-neighbour theorem gives the strictly sharper upper bound $O (n^{3 / 2})$ . Neither statement supplies a matching lower bound here, but the improvement already shows that the Erdős–Stone–Simonovits conclusion alone does not determine even the relevant subquadratic scale.

givenF4F5

2.1

Therefore the claimed universal determination is false. Erdős–Stone–Simonovits determines the leading quadratic density, not every lower-order extremal problem.

step 1.1step 1.2∎

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: 45 results over 21 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)