TheoremStatement: Literature-sourcedProof: AI-generatedprecheck 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.

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

Local pure or $x$ -sparse blockades yield a nice blockade

Statement

Let $ϵ \in (0, \frac{1}{2})$ and $d \geq 1$ , and put $x : = ϵ^{5 d}$ . Let $G$ be a graph with $∣ V (G) ∣ \geq ϵ^{- 10 d^{2}}$ . Assume that every induced subgraph $F$ of $G$ with $∣ V (F) ∣ \geq ϵ^{d} ∣ V (G) ∣$ has a pure or $x$ -sparse $(k, ∣ V (F) ∣ / k^{d})$ -blockade for some integer $k \in [2, x^{- 1}]$ . Then $G$ has a $(⌈ ϵ^{- 1} ⌉, x^{2 d} ∣ V (G) ∣)$ -blockade whose distinct block pairs are either complete or weakly $ϵ^{d}$ -sparse.

Facts & Assumptions

Given: The hypotheses in the statement.

[L1]

Theorem 6.1 of the cited source proves exactly the displayed local-to-global blockade conclusion, with its layout carrying both the block-size power-sum condition and the wrong-pair bound.

Proof

technique · translate the cited source theorem

1.1L1given

The cited source theorem applies to the hypotheses above and produces a blockade of length at least $ϵ^{- 1}$ , width at least $x^{2 d} ∣ V (G) ∣$ , and complete-or-weakly- $ϵ^{d}$ -sparse cross-pairs.

2.1step 1.1algebra∎

Since blockade length is integral, length at least $ϵ^{- 1}$ is equivalent to length at least $⌈ ϵ^{- 1} ⌉$ . This is exactly the stated conclusion.

Depends on

Used by

Property (*) and leaf reducibility imply generalized niceness Theorem
The five-vertex path is nice Theorem

Dependency tree · two levels

15 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 6.1 (standard reference, not scraped)

Local pure or xx-sparse blockades yield a nice blockade