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

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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.

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A sparse $P_{5}$ -free graph has an anticomplete two-blockade

Statement

There exists $η > 0$ such that every $η$ -sparse $P_{5}$ -free graph $G$ with $∣ V (G) ∣ \geq η^{- 1}$ contains an anticomplete blockade of length $2$ and width at least $η ∣ V (G) ∣$ .

Facts & Assumptions

Given: An $η$ -sparse $P_{5}$ -free graph $G$ with $∣ V (G) ∣ \geq η^{- 1}$ .

[L1]

Lemma 4.4 of the cited source proves exactly the displayed conclusion for $η = 2^{- 5}$ , using repeated large-component consequences of the failure of the desired anticomplete blockade.

Proof

technique · translate the cited source lemma

1.1

The cited source lemma assumes the negation of the desired blockade, first obtains a connected component of size at least $∣ V (G) ∣ / 2$ , and then obtains two further large connected pieces outside successive neighbourhoods.

L1given

2.1

The source shows that the absence of the blockade then forces five selected vertices to induce $P_{5}$ , contradicting the hypothesis. Therefore the stated anticomplete two-blockade exists.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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 4.4 (standard reference, not scraped)

A sparse P5P5​-free graph has an anticomplete two-blockade