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.
Why the self-density bound for a dense set carries a slack
The self-density counts ordered adjacent pairs of distinct vertices and divides by , so the diagonal pairs never contribute. A clique on vertices therefore has self-density , not . That is why the dense conclusion in A -sparse set has self-density at most , and a -dense set has self-density at least carries the reciprocal-size term. The density form of Rödl's theorem absorbs this term by changing its constants and states the cleaner threshold .
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Dependency tree · two levels
12 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
- Y. Zhao, Graph Theory and Additive Combinatorics, sec. 2.1 (standard reference, not scraped)