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.
Every perfect graph is 1-narrow
Statement
Every perfect finite graph is one-narrow.
Facts & Assumptions
Given: A perfect finite graph .
A graph is one-narrow when every good function on it has total weight at most (An -narrow graph).
A good function has weight at most on every perfect induced subgraph (An -narrow graph, A good function on a graph).
A perfect graph is one of its own perfect induced subgraphs (A perfect graph).
Proof
Let be a good function on . By [F3], the graph itself is a perfect induced subgraph of , so [F2] gives .
Since every good function has total weight at most , [F1] shows that is one-narrow.
Depends on
Used by
Nothing in the library uses this result yet.
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
- Maria Chudnovsky and Shmuel Safra, The Erdős-Hajnal conjecture for bull-free graphs, Section 2.1 (standard reference, not scraped)