A Sperner-type theorem for set-partition systems (Q2571313)

Let \(A\) be a set, \(| A| =n=ck.\) Further, let \(S\) be a system of partitions of \(A\) so that each partition in \(S\) consists of \(k\) classes, and if \(P_{1},P_{2}\) are two partitions in \(S\) with \(c_{1}\in P_{1},c_{2}\in P_{2},\) then \(c_{1}\) and \(c_{2}\) are incomparable sets. It is proved in the paper that \(S\) contains at most \(\binom{ck-1}{c-1}\) partitions.