Bose 1960 further results construction mutually orthogonal latin

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theorem_1: Bose, Shrikhande and Parker's main theorem: a pairwise balanced design of index unity on v treatments whose first l equiblock components form a clear set gives q* - 2 mutually orthogonal Latin squares of order v, where q* is the least of q_i + 1 over the clear components and q_i over the rest, given q_i - 1 mutually orthogonal Latin squares of each block size k_i.

theorem_10: Bose, Shrikhande and Parker's theorem that at least two orthogonal Latin squares of order v exist for every v > 6, so that among the orders v > 2 only 6 has no pair and Euler's conjecture fails for every v = 4t + 2 > 6.

theorem_8: Bose, Shrikhande and Parker's inequality that k <= N(m) + 1 gives N(km+1) >= min(N(k), N(k+1), 1+N(m)) - 1 and, for 1 < x < m, N(km+x) >= min(N(k), N(k+1), 1+N(m), 1+N(x)) - 1.

theorem_9: Bose, Shrikhande and Parker's method-of-differences construction of at least two orthogonal Latin squares of order 3m + 1 for every odd m, which with m = 4t + 3 covers every order 12t + 10.

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Bose, R. C. and Shrikhande, S. S. and Parker, E. T., Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler's conjecture. Canadian J. Math. 12 (1960), 189-203. doi:10.4153/cjm-1960-016-5.

Writing $N(v)$ for the maximum number of mutually orthogonal Latin squares of order $v$ and $n(v)$ for MacNeish's bound, the least of the prime powers $p_i^{n_i}$ in the prime-power decomposition of $v$, minus one, with $N(v)\ge n(v)$ (p. 189), the paper states three contributions (p. 190): (i) an improvement of the main theorem of Bose and Shrikhande's earlier memoir (its reference (6)), giving better bounds on $N(v)$; (ii) the method of differences, giving $N(v)\ge2$ for $v=14$, $26$ and $12t+10$; and (iii) the falsity of Euler's conjecture for all $v=4t+2>6$. The tools are pairwise balanced designs of index unity, BIB and group divisible designs, and orthogonal arrays. Theorem 1 (p. 191) turns a pairwise balanced design whose first $l$ equiblock components form a clear set into mutually orthogonal Latin squares. Theorems 2 and 3 (p. 193) derive bounds such as $N(v-1)\ge\min(N(k),1+N(k-1))-1$ from a BIB $(v;k)$; Theorems 4A, 4B and 4 (pp. 194--196) treat resolvable and separable BIB designs, for instance $N(v+r)\ge\min(N(k+1),1+N(r))-1$; Theorems 5--7 (pp. 196--197) give the analogues for group divisible designs $\mathrm{GD}(v;k,m;0,1)$, starting from $N(v)\ge\min(N(k),1+N(m))-1$; and Theorem 8 (p. 198) specialises them to $N(km+x)\ge\min(N(k),N(k+1),1+N(m),1+N(x))-1$ for $k\le N(m)+1$, $1<x<m$. Theorem 9 (p. 199) gives two orthogonal Latin squares of order $3m+1$ for odd $m$. Table I (p. 201) lists the orders $v\le154$ whose lower bound for $N(v)$ improves on Table I of reference (6); every bound listed exceeds $n(v)$. Theorem 10 (p. 202) gives two orthogonal Latin squares of every order $v>6$, and the paper concludes (p. 203) that among the orders $v>2$ only $6$ has no pair of orthogonal Latin squares.

Source: https://doi.org/10.4153/cjm-1960-016-5. No notice is printed (the running footer "Published online by Cambridge University Press" is not one); the journal's article page on Cambridge Core shows "Copyright © Canadian Mathematical Society 1960" and names no Creative Commons license (https://www.cambridge.org/core/product/identifier/S0008414X0000986X/type/journal_article, read 2026-10-02), every other right reserved.

Read status: claims checked for Theorems 1, 8, 9 and 10, Lemmas 3 and 4 and the statements of Theorems 2--7 (read clause by clause on the page images of the print, the constructions followed); the entries of Tables I and II and the printed squares were not checked. Nothing here is independently reviewed.

Results

• Theorem 1 (p. 191): a pairwise balanced design of index unity and type $(v;k_1,\ldots,k_m)$ whose components $(D_1),\ldots,(D_l)$, $l<m$, form a clear set, with $q_i-1$ mutually orthogonal Latin squares of order $k_i$, gives at least $q^*-2$ of order $v$, where $q^*=\min(q_1+1,\ldots,q_l+1,q_{l+1},\ldots,q_m)$.
• Theorem 8 (p. 198): if $k\le N(m)+1$, then $N(km+1)\ge\min(N(k),N(k+1),1+N(m))-1$ and, for $1<x<m$, $N(km+x)\ge\min(N(k),N(k+1),1+N(m),1+N(x))-1$.
• Theorem 9 (p. 199): at least two orthogonal Latin squares of order $3m+1$ for every odd $m$, hence of every order $12t+10$.
• Theorem 10 (p. 202; proof to p. 203): at least two orthogonal Latin squares of every order $v>6$.

Other statements, not given pages: Theorem 2 (p. 193), a BIB $(v;k)$ gives $N(v-1)\ge\min(N(k),1+N(k-1))-1$ and $N(v-x)\ge\min(N(k),N(k-1),1+N(k-x))-1$ for $2\le x\le k$; Theorem 4B (p. 195), a BIB $(v;k)$ with $r$ replications whose blocks split into sets of type I gives $N(v+r)\ge\min(N(k+1),1+N(r))-1$, which Example (7) (p. 195) applies to the symmetric BIB $(7;3)$ and the BIB $(57;8)$ to get $N(10)\ge2$ and $N(65)\ge7$; Theorem 5 (p. 196), a $\mathrm{GD}(v;k,m;0,1)$ gives $N(v)\ge\min(N(k),1+N(m))-1$; Lemma 4 (p. 202), $N(v)\ge2$ for $6<v\le726$.

Bears on. #724, whose $f(n)$ is this paper's $N(n)$ and which asks whether $f(n)\gg n^{1/2}$: Theorem 10 gives $N(n)\ge2$ for every $n>6$, and Table I gives bounds up to $N(65)\ge7$ for particular orders up to $154$; none of this concerns the growth of $N(n)$, and the paper does not answer the question. Its inequality Theorem 8 (ii) is the tool the Chowla--Erdős--Straus card records them using for $N(n)\to\infty$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.