Mixed orthogonal arrays, $k$-dimensional $M$-part Sperner multi-families, and full multi-transversals

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 $(m_1,m_2,...,m_M;L_1,L_2,...,L_M)$ 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 $i^{\text{th}}$ column. In this paper we define $k$-dimensional $M$-part Sperner multi-families 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 multi-families 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. 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 ith column. We extend the convex hull method to k-dimensional M-part Sperner multi-families, and allow additional conditions providing new results even for simple 1-part Sperner families.