A Sperner-type theorem (Q1068862)

Let \(P=P_ 1\times P_ 2\times...\times P_ M\) be the direct product of symmetric chain orders \(P_ 1,P_ 2,...,P_ M\). Let F be a subset of P containing no \(\ell +1\) elements which are identical in M-1 components and linearly ordered in the Mth one. Then max \(| F| \leq c.M^{1/2}.\ell.W(P)\), where W(P) is the cardinality of the largest level of P, and c is independent of P, M and \(\ell\). Infinitely many P show that this result is best possible for every M and apart from the constant factor c.