In a previous post, we saw that a card game with some particular properties is essentially the same thing as a finite projective plane (FPP). For some $n \ge 2$,

• Each card has $n+1$ symbols,
• Each symbol appears on $n+1$ cards,
• Any two cards share exactly one common symbol, and
• Any two symbols appear together on exactly one card.

$n$ is the order of the FPP. All manner of elusive combinatorial structures turn out to be equivalent to, constructible from, or otherwise intimately connected with FPPs. This post is about one particular case: complete sets of mutually orthogonal Latin squares.

Backtrack for a moment. A Latin square of order $n$ is an $n{\times}n$ grid containing the numbers $1,2,\ldots,n$ such that each row and each column contains all $n$ numbers. By the pigeonhole principle this is the same as saying no number appears twice in any row or any column. (If a number appears twice in a row, then there is not enough space left to fit the rest of the numbers in that row.) Here are three Latin squares with $n=4$: $$\mathcal{A} = \begin{bmatrix} 1&2&3&4 \\ 2&1&4&3 \\ 3&4&1&2 \\ 4&3&2&1 \end{bmatrix} \hspace{.5cm} \mathcal{B} = \begin{bmatrix} 1&2&3&4 \\ 3&4&1&2 \\ 4&3&2&1 \\ 2&1&4&3 \end{bmatrix} \hspace{.5cm} \mathcal{C} = \begin{bmatrix} 1&2&3&4 \\ 4&3&2&1 \\ 2&1&4&3 \\ 3&4&1&2 \end{bmatrix}$$

Latin squares are not remotely rare. In fact, they are easy to construct from blank grids: if you do one row at a time, you will never reach a dead end. But orthogonal Latin squares are harder to come by. $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ are not just any $4{\times}4$ Latin squares: together, they have the particularly elusive property that when any two are superimposed, every ordered pair of numbers is present. Here's $\mathcal{A}$ and $\mathcal{B}$ together:$$\begin{bmatrix}1,1&2,2&3,3&4,4 \\ 2,3&1,4&4,1&3,2 \\ 3,4&4,3&1,2&2,1 \\ 4,2&3,1&2,4&1,3\end{bmatrix}$$I.e. $\mathcal{A}$ and $\mathcal{B}$ are orthogonal. The same can be easily verified for $\mathcal{A}$ and $\mathcal{C}$, and for $\mathcal{B}$ and $\mathcal{C}$. Of course, by the pigeonhole principle, "every ordered pair of numbers appears once" is the same as saying "no ordered pair of numbers appears twice".

A natural* question to ask is how many mutually orthogonal Latin squares can be found.

Definition. Latin squares (of the same order) $\mathcal{A}_1,\ldots,\mathcal{A}_t$ are mutually orthogonal if for $i \ne j$, $\mathcal{A}_i$ and $\mathcal{A}_j$ are orthogonal.

Definition. $N(n)$ is the maximum size of a set of mutually orthogonal Latin squares (MOLS) of order $n$.

We have seen already that $N(4) \ge 3$. In fact, $N(4) = 3$.

Proposition. for $n \ge 2$, $N(n) \le n-1$.

Proof. Note that permuting the numbers of a Latin square (e.g. exchanging $1$s for $2$s and vice versa) preserves orthogonality. Therefore without loss of generality, each $\mathcal{A}_k$ has $[1~2~\cdots~n]$ in the first row.

Now look at, say, the second row** in the first column; let $s_k$ be the number there in $\mathcal{A}_k$. Every $s_k \ne 1$, since each $\mathcal{A}_k$ has a $1$ in the first row of the same column, and the number $1$ cannot appear twice in a column of a Latin square.

Furthermore, the $s_k$ are distinct, otherwise some ordered pair $s,s$ appears in this position and in the first row in the $s$th column, contradicting orthogonality.

So there are at most $n-1$ numbers which can appear in this position, and each $\mathcal{A}_k$ has a different number there. $\square$

The following definition is therefore fitting:

Definition. A set of MOLS of order $n \ge 2$ is complete if the size of the set is $n-1$.

So $\mathcal{A},\mathcal{B},\mathcal{C}$ are a complete set for $n=4$.

The main purpose of this post is to give a proof of the following fantastic theorem,*** due to R. C. Bose in 1938:

Theorem. There is a complete set of mutually orthogonal Latin squares of order $n$ if and only if there is a finite projective plane of order $n$.

The proof is constructive. We'll consider complete Card & Symbol Games (CSGs) with $n+1$ symbols per card, which for $n \ge 2$ are the same thing as FPPs, but the notation and intuition lends itself to a nice visual proof which accomplishes both the "if" and the "only if" in one fell swoop.

Before we attempt a general proof, it's worth trying an example. If we use our complete set $\mathcal{A},\mathcal{B},\mathcal{C}$ for $n=4$, there will be $4^2 + 4 + 1$ cards, each with $4+1$ symbols.

$5$ "special" cards are given below:

 $I$ $J$ $\Omega~\sigma_1~\sigma_2~\sigma_3~\sigma_4$ $\Omega~\tau_1~\tau_2~\tau_3~\tau_4$
 $z_{\mathcal{A}}$ $z_{\mathcal{B}}$ $z_{\mathcal{C}}$ $\Omega~\alpha_1~\alpha_2~\alpha_3~\alpha_4$ $\Omega~\beta_1~\beta_2~\beta_3~\beta_4$ $\Omega~\gamma_1~\gamma_2~\gamma_3~\gamma_4$

The other $4^2$ cards can be arranged in a $4{\times}4$ grid:

 $\sigma_1~\tau_1~\alpha_1~\beta_1~\gamma_1$ $\sigma_1~\tau_2~\alpha_2~\beta_2~\gamma_2$ $\sigma_1~\tau_3~\alpha_3~\beta_3~\gamma_3$ $\sigma_1~\tau_4~\alpha_4~\beta_4~\gamma_4$ $\sigma_2~\tau_1~\alpha_2~\beta_3~\gamma_4$ $\sigma_2~\tau_2~\alpha_1~\beta_4~\gamma_3$ $\sigma_2~\tau_3~\alpha_4~\beta_1~\gamma_2$ $\sigma_2~\tau_4~\alpha_3~\beta_2~\gamma_1$ $\sigma_3~\tau_1~\alpha_3~\beta_4~\gamma_2$ $\sigma_3~\tau_2~\alpha_4~\beta_3~\gamma_1$ $\sigma_3~\tau_3~\alpha_1~\beta_2~\gamma_4$ $\sigma_3~\tau_4~\alpha_2~\beta_1~\gamma_3$ $\sigma_4~\tau_1~\alpha_4~\beta_2~\gamma_3$ $\sigma_4~\tau_2~\alpha_3~\beta_1~\gamma_4$ $\sigma_4~\tau_3~\alpha_2~\beta_4~\gamma_1$ $\sigma_4~\tau_4~\alpha_1~\beta_3~\gamma_2$

It is slow but simple to verify that this is indeed a complete CSG. Notice:

• Each "cell" card must have one common symbol with each "special" card; since we can't have any more than $5$ cards with the symbol $\Omega$, this means each "cell" card must have exactly one $\sigma$, $\tau$, $\alpha$, $\beta$ and $\gamma$ symbol.
• The subscripts on the $\sigma$ and $\tau$ symbols are used to determine the row and column of the cell.
• The subscripts on the $\alpha$, $\beta$ and $\gamma$ symbols are given by the numbers in the Latin squares $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$.
• The "each number appears once in each row and column" rule for Latin squares, and the "each ordered pair appears once" rule for orthogonal Latin squares, both correspond with the "each pair of symbols appears together on one card" rule for complete CSGs.

The example contains almost all of the insight; the rest is formalism, except for the following lemma. Notice that e.g. the two cards $\newcommand{\card}[1]{\boxed{{\strut}~#1~}} \card{\sigma_2~\tau_3~\alpha_4~\beta_1~\gamma_2}$ and $\card{\sigma_4~\tau_2~\alpha_3~\beta_1~\gamma_4}$ have a common symbol $\beta_1$ - this is a requirement for a complete CSG - but do complete sets of MOLS always satisfy this? It means that given two grid positions not in the same row or column, there must be one Latin square from the complete set of MOLS with the same number in both positions.$$\begin{bmatrix} 1&2&3&4 \\ 2&1&\boxed{4}&3 \\ 3&4&1&2 \\ 4&\boxed{3}&2&1 \end{bmatrix} \hspace{.5cm} \begin{bmatrix} 1&2&3&4 \\ 3&4&\boxed{1}&2 \\ 4&3&2&1 \\ 2&\boxed{1}&4&3 \end{bmatrix} \hspace{.5cm} \begin{bmatrix} 1&2&3&4 \\ 4&3&\boxed{2}&1 \\ 2&1&4&3 \\ 3&\boxed{4}&1&2 \end{bmatrix}$$It's not obvious why this has to happen.

Lemma. If $\mathcal{A}_1,\ldots,\mathcal{A}_{n-1}$ is a complete set of MOLS, $i \ne i'$ and $j \ne j'$, then exactly one $\mathcal{A}_k$ has $(\mathcal{A}_k)_{i,j} = (\mathcal{A}_k)_{i',j'}$.

Proof. Without loss of generality, we can permute symbols so that every $(\mathcal{A}_k)_{i,j} = 1$. Let $l_k$ be the row containing the $1$ in column $j'$ of $\mathcal{A}_k$.

Since they are Latin squares, every $l_k \ne i$. Also, the $l_k$ are distinct: every pair of the Latin squares each has a $1$ in the same cell in column $j$, so by orthogonality they cannot also each have a $1$ in the same cell of column $j'$.

Therefore the $n-1$ distinct values of $l_k$ are each of the $n-1$ rows other than $i$, so one $l_k = i'$. $\square$

OK, now the proof of Bose's theorem. To avoid drowning in a sea of unnecessary variables, the subscript $-$ indicates an unknown number between $1$ and $n$.

Proof. Fix a distinguished symbol $\Omega$, and let $I = \card{\Omega~\sigma_1~\cdots~\sigma_n}$, $J = \card{\Omega~\tau_1~\cdots~\tau_n}$ and $n-1$ cards $z_{\mathcal{A}},z_{\mathcal{B}},\ldots,z_{\mathcal{H}}$ be the cards with symbol $\Omega$, with $z_{\mathcal{A}} = \card{\Omega~\alpha_1~\cdots~\alpha_n}$, $z_{\mathcal{B}} = \card{\Omega~\beta_1~\cdots~\beta_n}$, and so on up to $z_{\mathcal{H}} = \card{\Omega~\eta_1~\cdots~\eta_n}$. Denote these as the special cards.

Note that $\sigma_{-},\tau_{-},\alpha_{-},\ldots,\eta_{-}$ must be distinct: each can only appear once alongside $\Omega$, and there are $n+1$ such distinct cards.

Note that we're just choosing names, not imposing restrictions; the construction works either starting with a complete CSG, or starting from a blank slate in an attempt to construct one.

Proof (continued). We have accounted for $n+1$ cards and all $n^2 + n + 1$ symbols, so there are $n^2$ more cards; denote these as the cell cards. Any cell card $c$ must have exactly one symbol in common with each special card, and by assumption $c$ does not have the symbol $\Omega$. It follows that $c = \card{\sigma_{-}~\tau_{-}~\alpha_{-}~\cdots~\eta_{-}}$.

There must be $n$ cell cards with each $\sigma_{-}$, and $n$ cell cards with each $\tau_{-}$. Moreover, each ordered pair $\sigma_{-},\tau_{-}$ must appear on exactly one card. This gives an enumeration of the cell cards as $c_{i,j}$ for $1 \le i,j \le n$:

 $\sigma_1~\tau_1~\alpha_{-}~\cdots~\eta_{-}$ $\cdots$ $\sigma_1~\tau_n~\alpha_{-}~\cdots~\eta_{-}$ $\vdots$ $\ddots$ $\vdots$ $\sigma_n~\tau_1~\alpha_{-}~\cdots~\eta_{-}$ $\cdots$ $\sigma_n~\tau_n~\alpha_{-}~\cdots~\eta_{-}$

Now let $c_{i,j}$'s subscripts $\alpha_{-},\ldots,\eta_{-}$ be the numbers $\mathcal{A}_{i,j},\ldots,\mathcal{H}_{i,j}$ respectively: if the CSG is given, use the subscripts to determine the Latin squares, and vice versa.

We now check that from a complete set of MOLS, the resulting incidence structure is a complete CSG:

• By construction each card has $n+1$ symbols.
• $\Omega$ appears on $n+1$ cards. Each $\sigma_{-}$ and each $\tau_{-}$ appear on one special card and on $n$ cell cards, making $n+1$. Each $\alpha_{-}$ appears on one special card and $n$ cell cards since each number appears $n$ times in $\mathcal{A}$, making $n+1$; $\beta_{-},\ldots,\eta_{-}$ are similar.
• By construction each two special cards have exactly $\Omega$ in common, and each special card has one symbol in common with each cell card. Two cell cards with the same $\sigma_{-}$ do not have a common $\tau_{-}$ and vice versa, and either case has no common $\alpha_{-},\ldots,\eta_{-}$ since each number appears only once in each row and column of $\mathcal{A},\ldots,\mathcal{H}$. Two cell cards with the same $\alpha_{-}$ do not have a common $\beta_{-}$, as no ordered pair appears twice when $\mathcal{A}$ and $\mathcal{B}$ are superimposed; other pairs are similar. The tricky case of showing that two cell cards with no common $\sigma_{-}$ or $\tau_{-}$ do have a common symbol is handled by the lemma.
• Each two of the same $\sigma_{-},\tau_{-},\alpha_{-},\ldots,\eta_{-}$ appear together on one special card. Each $\sigma_{-},\tau_{-}$ appear together on one cell card by construction. Each $\sigma_{-},\alpha_{-}$ and each $\tau_{-},\alpha_{-}$ appear together on one cell card because each number appears once in each row and each column of $\mathcal{A}$; $\beta,\ldots,\eta$ are similar. Each $\alpha_{-},\beta_{-}$ appear together on one cell card because each ordered pair of numbers appears once when $\mathcal{A}$ and $\mathcal{B}$ are superimposed; other pairs are similar.

Next, we check that from a complete CSG, the result is a complete set of MOLS:

• Each $\sigma_{-}$ and $\alpha_{-}$ appear together on one card, so each number appears in one row of $\mathcal{A}$; columns and $\mathcal{B},\ldots,\mathcal{H}$ are similar. So the squares are indeed Latin.
• Each $\alpha_{-}$ and $\beta_{-}$ appear together on one card, so each ordered pair appears once when $\mathcal{A}$ and $\mathcal{B}$ are superimposed; other pairs are similar.
• The set of MOLS is complete since there are $n-1$ special cards $z_{\mathcal{A}},\ldots,z_{\mathcal{H}}$.

The result follows because for $n \ge 2$, a complete CSG is equivalent to a finite projective plane. $\square$

So as with FPPs, we can systematically construct a complete set of MOLS when $n$ is a power of a prime, it's impossible when $n=6$,**** and $n=12$ is an open problem.

*Mathematicians are greedy, so of course we want as many MOLS as possible.

**This proposition is often stated without the condition that $n \ge 2$; but the proof requires there to be a second row. Indeed, when $n=1$ there is a counterexample in the set containing the unique Latin square of order $1$, all pairs of distinct Latin squares from which are orthogonal.

***The theorem is well known to combinatorists, but proofs seem to be thin on the ground - even Wikipedia (as of writing) says "a set of n - 1 MOLS is equivalent to a finite projective plane of order n" with no citation.

****In fact, $N(6) = 1$.