A note on full transversals and mixed orthogonal arrays

We investigate a packing problem in M-dimensional grids, where bounds are given for the number of allowed entries in different axis-parallel directions. The concept is motivated from error correcting codes and from more-part Sperner theory. It is also closely related to orthogonal arrays. We prove that some packing always reaches the natural upper bound for its size, and even more, one can partition the grid into such packings, if a necessary divisibility condition holds. We pose some extremal problems on maximum size of packings, such that packings of that size always can be extended to meet the natural upper bound. 1 The concept of full transversals Let us be given positive integers n1, n2, . . . , nM and L1, L2, . . . , LM , such that L1 n1 + 1 ≤ L2 n2 + 1 ≤ · · · ≤ LM nM + 1 ≤ 1. (1) Let Π = [0, n1] × · · · × [0, nM ], and let I denote a subset of Π. We call an I ⊆ Π an (L1, L2, . . . , LM)-transversal, if there are no Li + 1 elements of I, any two of them differing from each other only in the i coordinate, for any i = 1, 2, . . . ,M . Identifying I with a 0-1 valued function defined on Π, namely the indicator function of I, an alternative definition of the (L1, L2, . . . , LM )-transversal is that for any k, fixing i1, . . . , îk, . . . , iM in an arbitrary fashion (where ̂ denotes a missing entry), nk ∑ ik=0 I(i1, . . . , ik, . . . , iM) ≤ Lk (2) holds. We talk about L-transversals, if L1 = L2 = · · · = LM = L. Lemma 1.1. If I is an (L1, . . . , LM)-transversal, then for every i = 1, . . . ,M |I| ≤ Li ni + 1 M ∏