@MISC{Dinitz13onthe, author = {J. H. Dinitz}, title = {On the existence of three dimensional Room frames and Howell cubes}, year = {2013} }

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Abstract

A Howell design of side s and order 2n+2, or more briefly an H(s, 2n+2) is an s × s array in which each cell is either empty or contains an unordered pair of elements from some 2n+2 set V such that (1) every element of V occurs in precisely one cell of each row and each column, and (2) every unordered pair of elements from V is in at most one cell of the array. It follows immediately from the definition of an H(s, 2n+2) that n+1 ≤ s ≤ 2n+1. A d-dimensional Howell design Hd(s, 2n + 2) is a d-dimensional array of side s such that (1) every cell is either empty or contains an unordered pair of elements from some 2n+ 2 set V, and (2) each two-dimensional projection is an H(s, 2n+ 2). The two boundary cases are well known designs: an Hd(2n + 1, 2n + 2) is a Room d-cube of side 2n+ 1 and the existence of d mutually orthogonal latin squares of order n + 1 implies the existence of an Hd(n + 1, 2n + 2). In this paper, we investigate the existence of Howell cubes, H3(s, 2n + 2). We completely determine the spectrum for H3(2n, 2n+ α) where α ∈ {2, 4, 6, 8}. In addition, we establish the existence of 3-dimensional Room frames of type 2v for all v ≥ 5 with only a few small possible exceptions for v. 1 1