Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01
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.
The -graph and co-
Definition
The -graph is the graph on vertices
with edge set
Thus is a five-vertex path and is a leaf attached to its middle vertex . The co- graph is the complement of this graph.
Used by
- The singleton Bird family has property (*) Corollary
- The singleton E-graph family is generalized nice Corollary
- The singleton family {E} has property (*) Corollary
- E overlap chains inside one comb block Definition
- A bipartite four-tooth comb has the co-E structural partition Example
- All E neighbourhood patterns under a complete nonedge pair Example
- The E theorem reaches an induced P₅ witness Example
- The E-graph and co-E by finite adjacency data Example
- The E-graph sits inside a one-subdivided star and deletes to the five-vertex path Example
- The no-E-copy boundary case of the comb partition Example
- The two induced co-E witnesses behind the forbidden path runs Example
- A mixed vertex on E is pure on both terminal edges Lemma
- Complete nonedge pairs force purity on induced E graphs Lemma
- E overlap classes form an anticonnected partition Lemma
- In a special-vertex comb of a co-E-free graph, vertices in other comb blocks remain pure to every H₅-overlap quotient block Lemma
- Relative to a complete nonedge pair in a co-E-free graph, a one-sided vertex mixed on an induced path avoids two consecutive nonneighbours and three consecutive neighbours Lemma
- Relative to a complete nonedge pair in a co-E-free graph, every one-sided vertex is pure to an induced H₅ Lemma
- The E-graph and Bird singleton families are leaf-reducible Lemma
- The E-graph and the Bird graph are wonderful Lemma
- The family consisting of H₅ and co-E has the Erdős–Hajnal property Lemma
- The pattern of the terminal H₅-overlap quotient is {H₅,co-E}-free Lemma
- The terminal E overlap pattern is E-free Lemma
- A special-vertex co-Bird-free comb admits an E-free structural partition Theorem
- A special-vertex comb in a co-E-free graph admits the {H₅,co-E} structural partition Theorem
- The E-graph has the Erdős-Hajnal property Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Figure 4 (standard reference, not scraped)