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.
Edge counts and densities between nonempty vertex sets
Definition
Let be a finite simple graph (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets) and let . Define the ordered cross-edge count When and are nonempty, their edge density is We omit the subscript when the graph is clear. If and are disjoint, this agrees with the edges between sets in Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs. If they overlap, an edge with both endpoints in contributes in both orientations, while diagonal pairs never contribute.
For a nonempty graph, write . By convention the density of the null graph is .
Depends on
Used by
- Three pair densities equal to 1/2 need not produce a single transversal triangle Counterexample
- The mean-square density, or energy, of a vertex partition Definition
- ε-regular pairs and self-regular vertex sets Definition
- The triangle counting lemma is exact for three complete cross-pairs Example
- Two complete pairs and one anticomplete pair produce exactly |X₁||X₂||X₃| induced copies of P₃ Example
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 11 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Y. Zhao, Graph Theory and Additive Combinatorics, Definition 2.1.1 (standard reference, not scraped)
- D. Conlon and J. Fox, Graph removal lemmas, sec. 2.1 (standard reference, not scraped)