The existence of Room 5-cubes
type: Item
instance of: scholarly article
title: The existence of Room 5-cubes
zbMATH Open document ID: 0638.05009
DOI: 10.1016/0097-3165(87)90049-5
author: J. H. Dinitz
published in: Journal of Combinatorial Theory. Series A
publication date: 1987-01-01T00:00:00Z
review text: A Room t-cube of side n is a t-dimensional array of side n on a set S of \(n+1\) symbols such that: (i) Each cell is either empty or contains an unordered pair of distinct symbols from S, (ii) Each symbol occurs in every (t-1)-dimensional flat in the array exactly once, (iii) Every unordered pair of symbols occurs precisely once in the array. It is known that the existence of a Room t-cube of side n is equivalent to the existence of t pairwise orthogonal symmetric Latin squares of side n. In this paper the author proves that if \(n\geq 11\) is odd (except possibly \(n=15)\), then there is a Room 5-cube of side n. The proof involves a recursive construction using 5-dimensional frames. The author also conjectures that there is a Room 5-cube of side 15.
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 4039920
zbMATH Keywords: one-factorizations
zbMATH Keywords: orthogonal Latin squares
zbMATH Keywords: Room t-cube
zbMATH Keywords: Room 5-cube
reviewed by: Gary L. Mullen
MaRDI profile type: Publication
full work available at URL: 0097-3165(87)90049-5
OpenAlex ID: W2032009881
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description: scientific article; zbMATH DE number 4039920