MathDB

Equality in Sperner

Source: Iranian National Olympiad (3rd Round) 2006

September 11, 2006

floor functioninequalitiesceiling functioncombinatorics proposedcombinatorics

Problem Statement

Let

A

be a family of subsets of

\{1,2,\ldots,n\}

such that no member of

A

is contained in another. Sperner’s Theorem states that

|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}

. Find all the families for which the equality holds.