DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-07-31

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 $A$ be a finite set. A family $F \subseteq [A]^{k}$ (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ) is intersecting if

$S \cap T \neq \emptyset$

for every $S, T \in F$ . For $k \geq 1$ and a fixed $a \in A$ , the star centred at $a$ is

$S_{a} : = {S \in [A]^{k} : a \in S} .$

Every star is intersecting, since all its members contain its centre.

Depends on

The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$

Used by

Dependency tree · two levels

10 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

Erdős-Ko-Rado theorem (Wikipedia) (standard reference, not scraped)