DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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 nice graph

Definition

Let $H$ be a finite graph. We say that $H$ is nice if there exists a real $d \geq 1$ such that for every $ϵ \in (0, \frac{1}{2})$ and every $\overline{H}$ -free graph $G$ with

$∣ V (G) ∣ \geq ϵ^{- 10 d^{2}},$

there is a $(⌈ ϵ^{- 1} ⌉, ϵ^{10 d^{2}} ∣ V (G) ∣)$ -blockade $B = (B_{1}, \dots, B_{ℓ})$ in $G$ such that for all distinct $i, j \in [ℓ]$ the pair $(B_{i}, B_{j})$ is either complete or weakly $ϵ^{d}$ -sparse.

The complement appears because the source proves niceness for $P_{5}$ through $\overline{P_{5}}$ -free graphs. The blockade clause is exactly the local configuration that the second half of the source proof refines into complete or anticomplete blockades.

Depends on

Used by

The five-vertex path is nice Theorem

Dependency tree · two levels

12 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, discussion before Lemma 3.4 (standard reference, not scraped)
Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, niceness discussion (standard reference, not scraped)