Subsquares and transversals in Latin squares. (English) Zbl 0718.05014

A Latin square without having an orthogonal mate is called a Bachelor square. For any n exhibit a Bachelor square of order n. The author claims that it is an old problem but no complete solution has been known. Euler proved that there are Bachelor squares of order 2n (e.g. the Cayley table of \(Z_{2n}\), the cyclic group of order 2n). The reviewer was not able to understand how far the author could get in the solution of the problem if the order of the Latin square is odd.
The author gives a reasonable conjecture. Namely \(\lim_{n\to \infty}\frac{B_ n}{-L_ n}=1\) where \(B_ n\) denotes the number of non isomorphic Bachelor squares of order n and \(L_ n\) is the number of nonisomorphic Latin squares of order n. At the end of the paper there are some results on the maximum number of sublatin squares of order 3.
{Reviewer’s remarks: 1) It is interesting to add that Euler’s result has at least three generalizations. Namely see Theorems 1.4.7 and 12.3.1 of the reviewer’s joint book with A. D. Keedwell [Latin squares and their applications (1974; Zbl 0283.05014)]. This book will be denoted by [DK]. E. Maillet in his paper “Sur les carrés latins d’Euler” in C. R. Assoc. France Av. Sci. 23, part 2, 244-251 (1984); see also pp. 445-447 of [DK], also generalized Euler’s result. 3) The author gives credit to a paper of E. T. Parker and L. Somer [Can. Math. Bull. 31, 409-413 (1988; Zbl 0669.05012)]. That paper had been anticipated by J. Bierbrauer [see e.g. Theorem 7.1 in Chapter 11 of Latin squares. New developments in the theory and applications, Edited by the reviewer and A. D. Keedwell (Amsterdam 1991)].}


05B15 Orthogonal arrays, Latin squares, Room squares