ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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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 $1\le k$ and $n\ge2k$ , and fix $a\in A$ . The star

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

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 $1\le k$ and $n\ge2k$ , 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 $1\le k$ and $n\ge 2k$ , 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 $\binom{n}{k} := \lvert [n]^{k}\rvert$ ).

Verification

technique · direct

1.1

Any two members of $\mathcal S_a$ intersect at $a$ , so the star is intersecting.

given

1.2

The map $S\mapsto S\setminus\{a\}$ is a bijection from $\mathcal S_a$ to the $(k-1)$ -subsets of $A\setminus\{a\}$ . Hence $|\mathcal S_a|=\binom{n-1}{k-1}$ .

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.

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Sources

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

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