Given a poset Q, a natural question to ask is “What is the largest subset of Q having no comparable elements?” In 1928 E. Sperner [SP] showed that for Q = B,, the set of all subsets of an n-set ordered by containment, the answer is Bn,,n,2,, its largest rank. Subsequently, sets of incomparable elements have become known as Sperner sets; the general maximization problem as the Sperner problem, and the property of a graded poset that the solution be the largest rank as the Sperner property. In 1970 G.-C. Rota [RO] asked if II,, the lattice of partitions of an n-set, ordered by refinement, has the Sperner property. E. R. Canfield showed in 1976 [CA] that the answer to Rota’s question is “not for n sufficiently large.” Follow-on papers by J. B. Shearer [SHI, J.C. Sha and D. J. Kleitman [SK] simplified Canfield’s argument and lowered the bound on II to 3.4 x 106. None of these studies, however, gave any additional information about the asymptotic growth of Sperner sets (e.g., can they grow significantly faster than the largest rank?). Meanwhile the present author [HA 1; HA 2; HA 31 developed a notion of morphism for the weighted Sperner problem, showing that Rota’s question is equivalent to that for the weighted poset II,/s,, = {a E 2: 1%~~ = n], partially ordered by the cone K, = (Zcq, j(6, + Sj ai+j)Jai,j 2 01, ai being 0 except in component i where it is 1, II,/S, has rank function r-(v) = Cy= iui, and weights w(u) = n!/IJ~,,(i!h,!. Since II,/s,, is embedded in R”, the possibility of passing to a continuous limit presented itself, the limit object being infinite dimensional (a