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The edge-density form of Rödl's theorem implies the maximum-degree form, with $ϵ$ and $δ$ each shrunk by a constant factor

Statement

Assume the edge-density form of Rödl's theorem holds at parameter $ϵ / 4$ with constant $δ_{0}$ . Then the maximum-degree form holds at parameter $ϵ$ with constant $δ_{0} / 2$ .

Facts & Assumptions

Given: A graph $H$ , a real $ϵ \in (0, \frac{1}{2})$ , and the edge-density form of Rödl's theorem at parameter $ϵ / 4$ with constant $δ_{0}$ .

[L1]

The edge-density form supplies, in every nonempty $H$ -free graph $G$ , a set $X$ of size at least $δ_{0} ∣ V (G) ∣$ with $d_{G} (X, X) \leq ϵ / 4$ or $d_{G} (X, X) \geq 1 - ϵ / 4$ (The edge-density form of Rödl's theorem: every nonempty $H$ -free graph has a linearly large set of self-density at most $ϵ$ or at least $1 - ϵ$ ).

[L2]

If $d_{G} (X, X) \leq c$ , then some subset $X^{'} \subseteq X$ with $∣ X^{'} ∣ \geq ∣ X ∣ / 2$ is $4 c$ -sparse (A set of self-density at most $c$ has a subset of at least half its size that is $4 c$ -sparse).

[L3]

A set is $c$ -dense in $G$ exactly when it is $c$ -sparse in $\overline{G}$ (A set is $c$ -sparse in $G$ exactly when it is $c$ -dense in $\overline{G}$ , so $c$ -restrictedness is complement-invariant).

Proof

technique · direct

1.1L1choose

Let $G$ be a nonempty $H$ -free graph. By [L1], choose $X \subseteq V (G)$ with $∣ X ∣ \geq δ_{0} ∣ V (G) ∣$ and either $d_{G} (X, X) \leq ϵ / 4$ or $d_{G} (X, X) \geq 1 - ϵ / 4$ .

2.1step 1.1L2

In the sparse branch, [L2] applied with $c = ϵ / 4$ gives a subset $X^{'} \subseteq X$ with $∣ X^{'} ∣ \geq ∣ X ∣ / 2$ that is $ϵ$ -sparse, hence $ϵ$ -restricted.

2.2step 1.1L2L3algebra

In the dense branch, the diagonal convention gives $d_{\overline{G}} (X, X) = 1 - 1 / ∣ X ∣ - d_{G} (X, X) \leq ϵ / 4$ ; applying [L2] to $\overline{G}$ yields a subset $X^{'} \subseteq X$ with $∣ X^{'} ∣ \geq ∣ X ∣ / 2$ that is $ϵ$ -sparse in $\overline{G}$ , and [L3] turns this into $ϵ$ -dense, hence $ϵ$ -restricted, in $G$ .

3.1step 2.1step 2.2∎

In either branch $∣ X^{'} ∣ \geq (δ_{0} / 2) ∣ V (G) ∣$ , so the maximum-degree form holds with constant $δ_{0} / 2$ .

Depends on

Used by

Bounded degree against bounded density: the two statements of Rödl's theorem, and which one is stronger Remark

Dependency tree · two levels

24 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

M. Chudnovsky, A. Scott, P. Seymour, and S. Spirkl, Strengthening Rödl's theorem, Theorems 1.1 and 1.2 (standard reference, not scraped)

The edge-density form of Rödl's theorem implies the maximum-degree form, with ϵϵ and δδ each shrunk by a constant factor