CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-07-31

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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.

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A symmetric chain decomposition gives a second proof of Sperner's bound

Statement

If $A$ has $n$ elements, every antichain in $B (A)$ has cardinality at most $(\binom{n}{⌊ n / 2 ⌋})$ .

Facts & Assumptions

Given: An $n$ -element set $A$ and an antichain $F$ in $B (A)$ .

[L1]

The Boolean lattice has a partition into symmetric chains (Every finite Boolean lattice has a symmetric chain decomposition).

[F1]

An antichain contains at most one element from any chain (Antichains, chain covers, and antichain covers of a poset).

[F2]

Rank $⌊ n / 2 ⌋$ consists of the $⌊ n / 2 ⌋$ -subsets of $A$ and has cardinality $(\binom{n}{⌊ n / 2 ⌋})$ (The Boolean lattice of subsets of a finite set and its rank levels, The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ).

Proof

technique · direct

1.1

Fix the symmetric saturated-chain decomposition supplied by [L1]. Every chain in it meets rank $⌊ n / 2 ⌋$ exactly once, because its consecutive ranks run from some $r \leq ⌊ n / 2 ⌋$ through $n - r \geq ⌊ n / 2 ⌋$ .

givenL1

2.1

Consequently the number of chains in the decomposition equals the cardinality of rank $⌊ n / 2 ⌋$ , hence equals $(\binom{n}{⌊ n / 2 ⌋})$ .

step 1.1F2

3.1

By [F1], the antichain $F$ contains at most one member from each chain. Its cardinality is therefore at most the number in step 2.1, which is Sperner's bound.

step 2.1F1∎

Depends on

Used by

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Sources

M. Keller and W. T. Trotter, Applied Combinatorics, §6.2 (standard reference, not scraped)