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.
The -graph and co- by finite adjacency data
Example
The -graph is a five-vertex path with a pendant edge at the middle vertex, and co- is its complement.
Facts & Assumptions
Given: The labelled graph with path vertices and an extra vertex .
The -graph has edge set , and co- is its complement (The -graph and co-, Graph isomorphisms, automorphisms and graph complements).
The standard path has consecutive edges and no others (Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices).
Verification
By [L1] and [L2], the vertices induce a , and the extra vertex is adjacent only to the middle vertex . So the labelled graph is exactly the -graph.
Taking the complement toggles each nonedge to an edge and each edge to a nonedge on the same six vertices. By [L1], that complement is co-.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
8 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
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Figure 4 (standard reference, not scraped)