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 sparse $\overline{P_{5}}$ -free graph yields a complete or anticomplete blockade or a sparser subgraph

Statement

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

a complete blockade of polynomial width;
an anticomplete blockade of polynomial width; or
a $c^{d}$ -sparse induced subgraph of linear size.

Facts & Assumptions

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

[L1]

Lemma 7.2 of the cited source proves the displayed complete-or-anticomplete blockade versus deeper-sparsification alternative with explicit parameters.

Proof

technique · translate the cited source lemma

1.1

The cited source lemma builds a maximal anticomplete blockade and verifies the size normalization needed before applying the preceding sparse trichotomy inside its last block.

L1given

2.1

Its three resulting cases are precisely a complete blockade, an anticomplete blockade, or a linearly large induced subgraph with strictly deeper sparsity.

step 1.1L1∎

Depends on

Used by

The minimal sparsity parameter drops below the target Lemma

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 7.2 (standard reference, not scraped)

A sparse P5‾P5​​-free graph yields a complete or anticomplete blockade or a sparser subgraph