ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 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.

All $k$ -sets through a fixed point form an intersecting family attaining the Erdős-Ko-Rado bound

Example

Let $A$ be an $n$ -element set with $1 \leq k$ and $n \geq 2 k$ , and fix $a \in A$ . The star

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

is intersecting and has cardinality $(\binom{n - 1}{k - 1})$ , attaining the Erdős-Ko-Rado bound.

Facts & Assumptions

Given: An $n$ -element set $A$ , natural numbers $1 \leq k$ and $n \geq 2 k$ , and a point $a \in A$ .

[L1]

Erdős-Ko-Rado bounds an intersecting family of $k$ -subsets by $(\binom{n - 1}{k - 1})$ and states that a star attains the bound (Erdős-Ko-Rado theorem: for $1 \leq k$ and $n \geq 2 k$ , an intersecting family of $k$ -subsets of an $n$ -set has size at most $(\binom{n - 1}{k - 1})$ , and a star attains the bound).

[F1]

$(\binom{m}{j})$ counts the $j$ -subsets of an $m$ -element set (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ).

Verification

technique · direct

1.1

Any two members of $S_{a}$ intersect at $a$ , so the star is intersecting.

given

1.2

The map $S \mapsto S ∖ {a}$ is a bijection from $S_{a}$ to the $(k - 1)$ -subsets of $A ∖ {a}$ . Hence $∣ S_{a} ∣ = (\binom{n - 1}{k - 1})$ .

givenF1

2.1

By [L1], step 1.2 equals the universal upper bound, so the star is extremal.

step 1.1step 1.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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)

All kk-sets through a fixed point form an intersecting family attaining the Erdős-Ko-Rado bound