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

TL;DR

This paper generalizes Sperner families using mixed orthogonal arrays, establishing new inequalities and conditions for homogeneity, and introduces constructions for complex array structures with applications to combinatorial design theory.

Contribution

It introduces k-dimensional M-part Sperner multi-families, proves multiple BLYM inequalities, and develops new constructions for mixed orthogonal arrays with specific constraints.

Findings

Homogeneity of multi-families under equality conditions for k<M

New BLYM inequalities for k-dimensional M-part Sperner multi-families

Extended convex hull method for Sperner families

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 $(m_1,m_2,...,m_M;L_1,L_2,...,L_M)$ 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.