The largest possible antichain of subsets of an n-element set, meaning a family in which no member is a subset of another, has size equal to the central binomial coefficient, achieved by taking all subsets of the middle size. Proved by Emanuel Sperner, it is a foundational result of extremal set theory, distinct from the same mathematician's combinatorial lemma on simplex triangulations.
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StatementSperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family. 1 Classification
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Source Sperner's Theorem (Wikipedia)
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1. Sperner's Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, opening sentence
Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family.
lead paragraph, sentence naming the publication year
It is named after Emanuel Sperner, who published it in 1928.
Proved By: Emanuel Sperner, Lead paragraph
It is named after Emanuel Sperner, who published it in 1928.
In Group: Ramsey Theory and Extremal Combinatorics, Lead section
in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family.
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