CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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When $k < n < 2 k$ , the entire $k$ th level is intersecting and exceeds the Erdős-Ko-Rado star bound

Statement refuted

The false statement False: the Erdős-Ko-Rado bound holds without the hypothesis $n \geq 2 k$ claims the Erdős-Ko-Rado star bound without assuming $n \geq 2 k$ .

Facts & Assumptions

Given: Natural numbers satisfying exactly $k < n < 2 k$ , an $n$ -element set $A$ , and the full level $F = [A]^{k}$ .

[F1]

An intersecting family has nonempty intersection between every pair of members, and $(\binom{n}{k}) = ∣ [A]^{k} ∣$ (Intersecting uniform families of finite sets, The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ).

[L1]

The binomial closed formula gives $(\binom{n}{k}) = \frac{n}{k} (\binom{n - 1}{k - 1})$ for $1 \leq k \leq n$ ( $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ ).

Counterexample

technique · direct

1.1

If $S, T \in [A]^{k}$ were disjoint, then $∣ S \cup T ∣ = 2 k > n = ∣ A ∣$ , impossible. Hence the entire level $F$ is intersecting.

givenF1

1.2

By [L1], $∣ F ∣ = (\binom{n}{k}) = (n / k) (\binom{n - 1}{k - 1})$ . Since $k < n$ , the factor $n / k$ is greater than $1$ , so $∣ F ∣ > (\binom{n - 1}{k - 1})$ .

givenF1L1algebra

2.1

Thus for every $k < n < 2 k$ , the full $k$ th level is an intersecting family larger than a star, refuting the bound outside its stated range.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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)

When k<n<2kk<n<2k, the entire kkth level is intersecting and exceeds the Erdős-Ko-Rado star bound