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The viral property implies the polynomial Rödl property

Statement

Every finite family of graphs with the viral property has the polynomial Rödl property.

Facts & Assumptions

Given: A finite family $F$ of graphs with the viral property.

[L1]

A family is viral when some exponent $d \geq 1$ makes the induced-copy implication hold for every $ϵ \in (0, \frac{1}{2})$ and every nonempty graph (The viral property for a finite forbidden family).

[L2]

If a graph is $F$ -free, then ${ind}_{H} (G) = 0$ for every $H \in F$ ( $H$ -free and $F$ -free graphs under the induced-subgraph convention, The induced-embedding count ${ind}_{H} (G)$ ).

[L3]

The polynomial Rödl property is the same restricted-set conclusion, but only for nonempty $F$ -free graphs (The polynomial Rödl property for a finite forbidden family).

Proof

technique · direct

1.1

Choose an exponent $d \geq 1$ witnessing the viral property of $F$ .

L1choose

1.2

Let $ϵ \in (0, \frac{1}{2})$ and let $G$ be a nonempty $F$ -free graph. Then [L2] gives ${ind}_{H} (G) = 0 < (ϵ^{d} ∣ V (G) ∣)^{∣ V (H) ∣}$ for every $H \in F$ .

L2algebra

2.1

Applying the viral implication from step 1.1 to the graph $G$ of step 1.2 yields an $ϵ$ -restricted vertex set of size at least $ϵ^{d} ∣ V (G) ∣$ . This is exactly the polynomial Rödl conclusion of [L3].

step 1.1step 1.2L1L3∎

Depends on

Used by

For a finite family, the Erdős–Hajnal property, the polynomial Rödl property, and virality are equivalent Theorem

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10 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

S. Huang, Y. Ju, and Y. Zhou, Erdős-Hajnal beyond the five-vertex path, §1.1 (standard reference, not scraped)