DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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.

Complete, anticomplete, pure, weakly sparse, and $x$ -sparse blockades

Definition

Let $B = (B_{1}, \dots, B_{t})$ be a blockade in a graph $G$ and let $x \in [0, 1]$ .

$B$ is complete when every pair $(B_{i}, B_{j})$ with $i < j$ is complete in the sense of Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs.
It is anticomplete when every pair $(B_{i}, B_{j})$ with $i < j$ is anticomplete.
It is pure when every pair $(B_{i}, B_{j})$ with $i < j$ is pure.
It is weakly $x$ -sparse when every pair $(B_{i}, B_{j})$ with $i < j$ is weakly $x$ -sparse in the sense of Sparsity of one vertex set to another, and weak sparsity of a pair.
It is $x$ -sparse when $B_{i}$ is $x$ -sparse to $B_{j}$ for every $i > j$ .

The last condition depends on the order of the blocks, while the first four do not.

Depends on

Used by

Sparse orientations of a blockade Definition
The pattern graph of a pure blockade Definition
A pure blockade that is neither complete nor anticomplete Example
FALSE: every pure blockade is either complete or anticomplete False statement
FALSE: reversing the order of the blocks never changes x-sparsity False statement
A maximal pure blockade with large total a-mass must already have at least ε⁻² blocks Lemma
Complete or anticomplete blockade hypotheses force an ε-restricted induced subgraph Theorem

Dependency tree · two levels

7 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 (standard reference, not scraped)
Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path (standard reference, not scraped)

Complete, anticomplete, pure, weakly sparse, and xx-sparse blockades