How 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.
Intersecting uniform families of finite sets
Definition
Let be a finite set. A family (The set of -element subsets and the binomial coefficient ) is intersecting if
for every . For and a fixed , the star centred at is
Every star is intersecting, since all its members contain its centre.
Depends on
Used by
- When k<n<2k, the entire kth level is intersecting and exceeds the Erdős-Ko-Rado star bound Counterexample
- At most k cyclic intervals of length k in a cyclic order are pairwise intersecting when the ground-set size is at least 2k Lemma
- Erdős-Ko-Rado theorem: for 1≤ k and n≥ 2k, an intersecting family of k-subsets of an n-set has size at most binomn-1k-1, and a star attains the bound Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 37 results over 17 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
- Erdős-Ko-Rado theorem (Wikipedia) (standard reference, not scraped)