Alphabeta MathHow 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.
An -narrow graph
Definition
Let be real. A finite graph is -narrow when every good function on satisfies
(A good function on a graph, Real powers for positive bases, with the zero-base positive-exponent convention).
In particular, a graph is one-narrow when every good function has total weight at most , and it is two-narrow when every good function has sum of squares at most . The source's word narrow corresponds to two-narrow in this notation.
Depends on
Used by
- The five-cycle is 2-narrow but not 1-narrow Example
- Every perfect graph is 1-narrow Proposition
- An α-narrow graph contains a perfect induced subgraph of order at least |V(G)|^1/α Theorem
- Every basic bull-free graph is 2-narrow Theorem
- Substituting two α-narrow graphs yields another α-narrow graph Theorem
Dependency tree · two levels
5 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
- Maria Chudnovsky, The Erdős-Hajnal Conjecture: A Survey, Section 2 (standard reference, not scraped)
- Maria Chudnovsky and Shmuel Safra, The Erdős-Hajnal conjecture for bull-free graphs, Section 1 (standard reference, not scraped)