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.

No vertex is mixed on many blocks of a semisparse blockade

Statement

There exist constants $c > 0$ and $d \geq 1$ such that the following holds. Let $G$ be a sufficiently large $c$ -sparse $\overline{P_{5}}$ -free graph, and let $(B_{1}, \dots, B_{t})$ be a blockade from outcome 2 of A semisparse blockade can be sampled to anticonnected blocks with nearly pure relations. Then at least one of the following holds:

$G$ has a $c^{d}$ -sparse induced subgraph of linear size; or
every vertex outside the blockade is mixed on fewer than $c t$ blocks, where a vertex $v$ is mixed on $B_{i}$ when the pair $({v}, B_{i})$ is mixed in the sense of Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs.

Facts & Assumptions

Given: A graph $G$ and a blockade $(B_{1}, \dots, B_{t})$ as in the statement.

[L1]

Claim 7.1.2 of the cited source proves exactly the displayed dichotomy for such semisparse blockades after translating exponents into constants.

Proof

technique · translate the cited source claim

1.1

The cited source claim proves exactly this mixed-block dichotomy after translating exponents into constants.

L1given

2.1

Therefore one of the two displayed outcomes holds.

step 1.1∎

Depends on

Used by

A sparse P₅-free graph yields deeper sparsification or a complete blockade or a large anticomplete set Lemma

Dependency tree · two levels

16 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 7.1.2 (standard reference, not scraped)