Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.
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Blockades, their length, their width, and their support
Definition
Let be a finite graph, and let be real with and . An -blockade in is a sequence
of pairwise disjoint nonempty subsets of such that and for every .
Since the actual length is an integer, a real lower bound is equivalent to . Thus this notation includes the usual integer length parameters while also allowing the real thresholds, such as , used in blockade estimates.
Each is a block. The length of is , its width is
and its support is
Used by
- Large almost-pure pair hypotheses yield a complete or anticomplete blockade Corollary
- A blockade-rainbow induced copy Definition
- A nice graph Definition
- Coherent graphs and support-regular blockades Definition
- Combs in a graph Definition
- Complete, anticomplete, pure, weakly sparse, and x-sparse blockades Definition
- Generalized nice finite graph families Definition
- Integral geometric layers of a decreasing block partition Definition
- Labelled blowup and good induced copy Definition
- Property (*) for a finite graph family Definition
- Qid restricted blockade with empty blocks Definition
- Quantitative divisiveness for a finite forbidden family Definition
- Sparse orientations of a blockade Definition
- The mixed-block reachability relation on a blockade Definition
- Wonderful finite graph families Definition
- A four-block blockade-rainbow copy of P₄ Example
- A large almost-pure pair extends an anticomplete blockade Example
- A sparse P₅-free graph with an anticomplete two-blockade Example
- A three-block blockade and its width Example
- Thinning a four-block weakly sparse blockade to directional sparse subblocks Example
- Two large anticonnected components give a complete two-blockade Example
- A complete-or-weakly-sparse blockade can be thinned to equal subblocks with directional sparsity Lemma
- A complete-or-weakly-sparse blockade yields a complete subblockade or an anticonnected thinning Lemma
- A maximal layout has at most ε⁻¹ blocks Lemma
- A semisparse blockade can be sampled to anticonnected blocks with nearly pure relations Lemma
- A sparse host has many leaf extensions, few smaller copies, or a long sparse blockade Lemma
- A sparse P₅-free graph yields a complete or anticomplete blockade or a sparser subgraph Lemma
- A transversal of wide structural blocks yields the pure blockade outcome Lemma
- A wonderful anticonnected complete-or-sparse blockade yields a restricted subgraph or a large anticomplete pair Lemma
- An iterative sparsification step for sparse P₅-free graphs Lemma
- An x-sparse blockade iteration yields further sparsification or a pure blockade Lemma
- Homogeneous sets in pure-blockade patterns lift to complete or anticomplete blockades Lemma
- Iterated mixed quotients of an H₅-overlap blockade terminate at a pure blockade Lemma
- Quantitatively divisive finite families are viral Lemma
- Small anticonnected components yield a complete blockade Lemma
- A long blockade without a large pure pair contains a rainbow forest or its complement Theorem
- A long blockade yields a wide cograph-pattern subblockade or a rainbow forest Theorem
- A pure blockade with a perfect pattern has a large complete or anticomplete subblockade Theorem
- A tau-critical graph has no wide pure blockade with cograph pattern Theorem
- A wide coherent blockade contains a blockade-rainbow copy of a forest Theorem
…and 4 more results.
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- T. H. Nguyen, Notes on Recent Work on the Erdős–Hajnal Conjecture (standard reference, not scraped)
- Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. VII. The five-vertex path (standard reference, not scraped)