TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-31

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The five-vertex path has the polynomial Rödl property

Statement

The graph $P_{5}$ has the polynomial Rödl property.

Facts & Assumptions

Given: The graph $P_{5}$ .

[L1]

There exists $a \geq 1$ such that every $\overline{P_{5}}$ -free graph either has an $x$ -restricted induced subgraph of size at least $x^{a} ∣ V (G) ∣$ or has a complete or anticomplete $(k, ∣ V (G) ∣ / k^{a})$ -blockade for some $k \in [2, x^{- 1}]$ ( $\overline{P_{5}}$ -free graphs yield a polynomial restricted set or a complete or anticomplete blockade).

[L2]

The blockade alternative alone already forces an $x$ -restricted induced subgraph of size at least $x^{3 a} ∣ V (G) ∣$ (Complete or anticomplete blockade hypotheses force an $ϵ$ -restricted induced subgraph).

Proof

technique · direct

1.1

Let $a$ be as in [L1]. Fix $x \in (0, \frac{1}{2})$ and a $P_{5}$ -free graph $G$ . Then $\overline{G}$ is $\overline{P_{5}}$ -free. Apply [L1] to every induced subgraph $F$ of $\overline{G}$ with $∣ V (F) ∣ \geq x^{2 a} ∣ V (G) ∣$ . If any such $F$ has an $x$ -restricted set of size at least $x^{a} ∣ V (F) ∣$ , that set has size at least $x^{3 a} ∣ V (G) ∣$ and is also $x$ -restricted in $G$ .

L1givencasesalgebra

2.1

Otherwise every such $F$ has the complete-or-anticomplete blockade supplied by [L1], so the hypothesis of [L2] holds for $\overline{G}$ . Applying [L2] yields an $x$ -restricted induced subgraph on at least $x^{3 a} ∣ V (G) ∣$ vertices; the same vertex set is $x$ -restricted in $G$ .

step 1.1L2cases

3.1

Therefore $P_{5}$ has the polynomial Rödl property.

step 2.1∎

Depends on

Used by

The five-vertex path and its complement have the Erdős-Hajnal property Corollary

Dependency tree · two levels

12 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

Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. VII. The five-vertex path, Theorem 1.5 (standard reference, not scraped)