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

1.1

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.

L1given

2.1

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

step 1.1L1∎

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