Mixed Orthogonal Arrays, k-Dimensional M-Part Sperner Multifamilies, and Full Multitransversals

Abstract

Aydinian et al. [J. Combinatorial Theory A 118(2)(2011), 702–725] substituted the usual BLYM inequality for L-Sperner families with a set of M inequalities for (m₁,m₂,…,m _M ; L₁,L₂,…,L _M ) type M-part Sperner families and showed that if all inequalities hold with equality, then the family is homogeneous. Aydinian et al. [Australasian J. Comb. 48(2010), 133–141] observed that all inequalities hold with equality if and only if the transversal of the Sperner family corresponds to a simple mixed orthogonal array with constraint M, strength M − 1, using m _i  + 1 symbols in the i^th column. In this paper we define k-dimensional M-part Sperner multifamilies with parameters $L_P:~P\in\binom{[M]}{k}$ and prove $\binom{M}{k}$ BLYM inequalities for them. We show that if k < M and all inequalities hold with equality, then these multifamilies must be homogeneous with profile matrices that are strength M − k mixed orthogonal arrays. For k = M, homogeneity is not always true, but some necessary conditions are given for certain simple families. These results extend to products of posets which have the strong normalized matching property. Following the methods of Aydinian et al. [Australasian J. Comb. 48(2010), 133–141], we give new constructions to simple mixed orthogonal arrays with constraint M, strength M − k, using m _i  + 1 symbols in the i^th column. We extend the convex hull method to k-dimensional M-part Sperner multifamilies, and allow additional conditions providing new results even for simple 1-part Sperner families.