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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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An $x$ -sparse blockade iteration yields further sparsification or a pure blockade

Statement

Let $x \in (0, \frac{1}{2})$ . There exist constants $c > 0$ and $d \geq 1$ such that the following holds. If $G$ is a sparse $\overline{P_{5}}$ -free graph and $B = (B_{0}, \dots, B_{n})$ is a maximal $x$ -sparse blockade whose last block still has linear size, then either:

some induced subgraph of $B_{n}$ is substantially sparser than $G$ ; or
$G$ has a pure blockade of polynomial width.

Facts & Assumptions

Given: A sparse $\overline{P_{5}}$ -free graph $G$ and a maximal $x$ -sparse blockade $B = (B_{0}, \dots, B_{n})$ with large last block $B_{n}$ .

[L1]

Lemma 5.3 of the cited source proves the stated maximal-blockade alternative with explicit constants, sizes, and exponents.

Proof

technique · translate the cited source lemma

1.1

The cited source lemma applies the preceding sparse-pair trichotomy to the final block and uses the correctly oriented sparse relations to extend the blockade whenever the sparse-pair outcome occurs.

L1given

2.1

Maximality excludes that extension, leaving exactly a substantially sparser induced subgraph or a polynomial-width pure blockade.

step 1.1L1∎

Depends on

Used by

An iterative sparsification step for sparse P₅-free graphs Lemma

Dependency tree · two levels

14 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 5.3 (standard reference, not scraped)

An xx-sparse blockade iteration yields further sparsification or a pure blockade