TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$\overline{P_{5}}$ -free graphs yield a polynomial restricted set or a complete or anticomplete blockade

Statement

There exists $a \geq 1$ such that for every $x \in (0, \frac{1}{2})$ and every $\overline{P_{5}}$ -free graph $G$ , at least one of the following holds:

$G$ has an $x$ -restricted induced subgraph with at least $x^{a} ∣ V (G) ∣$ vertices; or
$G$ has a complete or anticomplete $(k, ∣ V (G) ∣ / k^{a})$ -blockade for some integer $k \in [2, x^{- 1}]$ .

Facts & Assumptions

Given: A parameter $x \in (0, \frac{1}{2})$ and a $\overline{P_{5}}$ -free graph $G$ .

[L1]

Lemma 7.3 of the cited source proves exactly the displayed restricted-set or complete/anticomplete-blockade alternative, with explicit constants.

Proof

technique · direct from the minimal-threshold alternative

1.1L1given

The cited source lemma runs the minimal-threshold argument with all size and exponent bounds and proves the first alternative whenever the blockade alternative is absent.

2.1step 1.1L1∎

Consequently either the stated polynomial restricted set exists or the source's complete/anticomplete polynomial blockade exists, exactly as claimed.

Depends on

Used by

The five-vertex path has the polynomial Rödl property Theorem

Dependency tree · two levels

8 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, Lemma 7.3 (standard reference, not scraped)

P5‾P5​​-free graphs yield a polynomial restricted set or a complete or anticomplete blockade