LemmaStatement: 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.

A semisparse blockade can be sampled to anticonnected blocks with nearly pure relations

Statement

There exist constants $c > 0$ and $d \geq 1$ such that every sufficiently large sparse $\overline{P_{5}}$ -free graph has at least one of the following:

a complete blockade of linear width; or
a blockade $(B_{1}, \dots, B_{t})$ with $t \geq c^{- 1}$ , every block of size at least $c ∣ V (G) ∣$ , each $B_{i}$ anticonnected, and every pair $(B_{i}, B_{j})$ with $i \neq j$ either complete or weakly $c$ -sparse.

Facts & Assumptions

Given: A sufficiently large sparse $\overline{P_{5}}$ -free graph $G$ .

[L1]

Claim 7.1.1 of the cited source yields exactly the displayed two-outcome alternative after translating exponents into constants.

Proof

technique · translate the cited source claim

1.1

The cited source claim yields exactly this semisparse-blockade or complete-blockade alternative after translating exponents into constants.

L1given

2.1

Therefore one of the two displayed outcomes holds.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

19 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, Claim 7.1.1 (standard reference, not scraped)