Wilson 1974 number mutually orthogonal latin squares

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theorem_2_3: Wilson's product-type inequality for mutually orthogonal Latin squares, which drops the hypothesis m <= N(t)+1 from the Bose-Shrikhande-Parker inequality and drives the paper's bounds N(n) >= 2 and N(n) >= n^{1/17} - 2.

theorem_3_1: Wilson's short proof of the Bose-Shrikhande-Parker theorem that a pair of orthogonal Latin squares of order n exists for every n other than 2 and 6.

theorem_4_2: Wilson's lower bound for the largest number N(n) of mutually orthogonal Latin squares of order n: beyond the constant n_0 of Lemma 4.1, N(n) is at least n to the power 1/17, minus 2.

theorem_5_1: Wilson's bound that six mutually orthogonal Latin squares of order n exist for every n above 90, obtained from his inequalities with m = 7.

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R. M. Wilson, Concerning the number of mutually orthogonal Latin squares, Discrete Math. 9 (1974), 181--198 (received 13 March 1973), DOI 10.1016/0012-365X(74)90148-4.

The copy read for this card is a scan of the eighteen printed pages (head "DISCRETE MATHEMATICS 9 (1974) 181-198"; physical PDF p. $n$ is printed p. $180+n$) with an OCR text layer that garbles many formulas. The statements below were checked on the page images. Provenance: obtained in the repository's survey download of September 2026; the download URL was not recorded; 1,378,845 bytes. The scan prints "DISCRETE MATHEMATICS 9 (1974) 181-198. © North-Holland Publishing Company" at the head of its first page (the text layer renders the symbol as "o"), every other right reserved.

Read status: claims checked for Theorems 1.1--1.5, 2.3, 3.1, 4.2 and 5.1 and Lemma 4.1 (statements read clause by clause, with the exponent $1/17$ confirmed by the abstract and by the inequalities of the proof); the proofs and the construction of section 2 were not checked.

Contents

• Definitions (pp. 181--182): Latin squares of order $n$ as maps $R\times C\to S$, orthogonality, and $N(n)$, the largest size of a set of mutually orthogonal Latin squares of order $n$. Theorems 1.1--1.4 (p. 182): $1\le N(n)\le n-1$ for $n\ge2$; $N(n)=n-1$ for prime powers; $N(nm)\ge\min\{N(n),N(m)\}$; hence the MacNeish--Mann bound $N(n)\ge\min_i(p_i^{a_i}-1)$ over the prime-power factorization of $n$.
• History (pp. 182--183): Euler's and MacNeish's conjectures, Tarry's $N(6)=1$, the counterexamples of Parker and of Bose and Shrikhande; Theorem 1.5 (Bose, Shrikhande and Parker; p. 183): if $m\le N(t)+1$ and $1<u<t$, then $N(mt+u)\ge\min\{N(m)-1,N(m+1)-1,N(t),N(u)\}$. Chowla, Erdős and Straus (1960) noted that $N(n)\to\infty$ follows from Theorems 1.4 and 1.5, and proved $N(n)>\frac13n^{1/91}$ for large $n$ by Brun's sieve; Rogers (1964) obtained $N(n)>n^{1/(42+\epsilon)}$ for $n>n_\epsilon$ using Buchstab's result; Hanani proved $N(n)\ge3$ for $n>51$, $N(n)\ge5$ for $n>62$ and $N(n)\ge29$ for $n>34{,}115{,}553$.
• Section 2 (pp. 183--187): transversal designs $\mathrm{TD}(k,n)$, the construction Theorem 2.2 (p. 184) and its corollaries; Theorem 2.3 (p. 186): if $0\le u\le t$, then $N(mt+u)\ge\min\{N(m),N(m+1),N(t)-1,N(u)\}$. This is the replacement for Theorem 1.5 without the hypothesis $m\le N(t)+1$ that p. 183 announces; its minimum has $N(m)$ and $N(m+1)$ in place of $N(m)-1$ and $N(m+1)-1$, and $N(t)-1$ in place of $N(t)$.
• Theorem 3.1 (p. 187): $N(n)\ge2$ for $n\ne2,6$, the Bose, Shrikhande and Parker theorem, proved again on pp. 187--189: orders $10$ and $14$ are taken from their paper [3], Theorem 1.4 covers $n\not\equiv2\pmod 4$, and Theorem 2.3 with $m=3$ covers $n\equiv2\pmod 4$, $n\ge18$.
• Theorem 4.2 (p. 194; proof to p. 195): for $n>n_0$, $N(n)\ge n^{1/17}-2$. The proof writes $n=mt+u$ with $0<u<t$, choosing $m$, $t$ and $u$ through Buchstab's sieve result (Lemma 4.1) so that Theorem 1.4 gives $N(m),N(m+1)\ge n^{1/17}-2$ and $N(t),N(u)\ge n^{1/17}-1$, and then applies Theorem 2.3; Remark 4.3 says the "$-2$" can be removed with more of Buchstab's result.
• Theorem 5.1 (p. 196): $N(n)\ge6$ whenever $n>90$, from Theorems 2.4--2.5 with $m=7$ and the consecutive prime powers $7,8,9$.

Results

• Theorem 2.3 (p. 186): if $0\le u\le t$, then $N(mt+u)\ge\min\{N(m),N(m+1),N(t)-1,N(u)\}$.
• Theorem 3.1 (p. 187; proof to p. 189): $N(n)\ge2$ for $n\ne2,6$.
• Theorem 4.2 (p. 194; proof to p. 195): for $n>n_0$, $N(n)\ge n^{1/17}-2$, with $n_0$ the constant of Lemma 4.1 (p. 193).
• Theorem 5.1 (p. 196; proof to p. 197): $N(n)\ge6$ whenever $n>90$.

Compiled scope

Pages 181--183 were read in full, and sections 2--5 for their statements and the outlines of the proofs of Theorems 3.1 and 4.2, all on the page images; no proof was checked. Nothing here is independently reviewed.

Bears on. #724, whose $f(n)$ is this paper's $N(n)$ and which asks whether $f(n)\gg n^{1/2}$: Theorem 4.2 gives $N(n)\ge n^{1/17}-2$ for $n>n_0$, a lower bound of smaller order that does not answer the question, proved through Theorem 2.3. The introduction records the earlier sieve bounds $N(n)>\frac13n^{1/91}$ for sufficiently large $n$, of [CES60], and $N(n)>n^{1/(42+\epsilon)}$ for $n>n_\epsilon$, of Rogers.

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