TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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

Statement

Let $F$ be a finite family of graphs. The following are equivalent:

$F$ has the Erdős–Hajnal property.
$F$ has the polynomial Rödl property.
$F$ is viral.

Facts & Assumptions

Given: A finite family $F$ of graphs.

[L1]

Every finite family with the Erdős–Hajnal property is viral (Every finite family with the Erdős–Hajnal property is viral).

[L2]

Every viral finite family has the polynomial Rödl property (The viral property implies the polynomial Rödl property).

[L3]

Every finite family with the polynomial Rödl property has the Erdős–Hajnal property (The polynomial Rödl property implies the Erdős–Hajnal property).

Proof

technique · direct

1.1

Assertion 1 implies assertion 3 by [L1].

1.2

Assertion 3 implies assertion 2 by [L2].

1.3

Assertion 2 implies assertion 1 by [L3].

2.1

The three implications close the cycle $1 \Rightarrow 3 \Rightarrow 2 \Rightarrow 1$ , so the three assertions are equivalent.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Dependency tree · two levels

29 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, Theorem 1.3 (standard reference, not scraped)