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Erdős–Stone for balanced blowups: $π (K_{r} [s]) = 1 - 1 / (r - 1)$ for $r \geq 2$

Statement

For integers $r \geq 2$ and $s \geq 1$ ,

$π (K_{r} [s]) = 1 - \frac{1}{r - 1} .$

Equivalently,

$ex (n, K_{r} [s]) = (1 - \frac{1}{r - 1} + o (1)) (\binom{n}{2}) .$

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

For $n \in N$ and $r \geq 1$ , Turán's theorem gives $ex (n, K_{r + 1}) = e (T_{n, r})$ , and an $n$ -vertex $K_{r + 1}$ -free graph attains equality exactly when it is isomorphic to $T_{n, r}$ (Turán's theorem with equality: $ex (n, K_{r + 1}) = e (T_{n, r})$ , and $T_{n, r}$ is the unique extremal graph).

[F2]

For every $ε > 0$ , a sufficiently large graph with density at least $π (H) + ε$ contains at least $δ n^{v (H)}$ injective copies of $H$ (Above Turán density, a graph contains a positive-density family of copies of the forbidden graph).

[F3]

For fixed integers $r \geq 2$ and $s \geq 2$ , $ex (n, K_{s, \dots, s}^{(r)}) = O_{r, s} (n^{r - 1 / s^{r - 1}}) = o (n^{r})$ (Hypergraph KST: $ex (n, K_{s, \dots, s}^{(r)}) = O_{r, s} (n^{r - 1 / s^{r - 1}}) = o (n^{r})$ ).

[F4]

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 $H [s]$ ).

Proof

technique · turn many cliques into a complete partite clique hypergraph

1.1

If $s = 1$ , the assertion is exactly Turán's theorem for $K_{r}$ . Assume $s \geq 2$ . The graph $T_{n, r - 1}$ contains no $K_{r}$ , hence no $K_{r} [s]$ , and its normalized edge count tends to $1 - 1 / (r - 1)$ . This gives the lower bound for the density.

givenF1F4

1.2

Fix $ε > 0$ . A graph with density at least $1 - 1 / (r - 1) + ε$ has, by Turán's theorem and supersaturation for $K_{r}$ , at least $c n^{r}$ injective embeddings of $K_{r}$ for all large $n$ , for some $c > 0$ . Each clique supports at most $r!$ such embeddings, so after decreasing $c$ there are at least $c n^{r}$ distinct $r$ -vertex cliques. Make these clique vertex sets the edges of an $r$ -uniform hypergraph.

givenF1F2

2.1

Hypergraph KST says that, for large $n$ , an $r$ -graph with $c n^{r}$ edges contains $K_{s, \dots, s}^{(r)}$ . In the underlying graph every transversal of its $r$ parts is a clique. Given vertices in two distinct parts, extend them by one vertex from each other part; the resulting clique shows their cross edge is present. Thus the original graph contains $K_{r} [s]$ .

step 1.2givenF3F4

3.1

Step 2.1 gives the density upper bound $1 - 1 / (r - 1) + ε$ for every $ε > 0$ , while step 1.1 gives the matching lower bound. Hence the limit and the equivalent asymptotic formula follow, including $s = 1$ .

step 1.1step 2.1∎

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: 39 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)
Reinhard Diestel, Graph Theory, Chapter 7 (standard reference, not scraped)

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