Re-instate Example 5.4

[MHenderson]

I don't think smallmatrix is going to save us here.

[MHenderson]

\begin{example}
Consider the following $\BRS(8)$.
\begin{equation}
  \begin{bmatrix}
    \infty 0 &    26    &    45    &            &    13    &            &          \\
             & \infty 1 &    30    &     56     &          &     24     &          \\
             &          & \infty 2 &     41     &    60    &            &    35    \\
      46     &          &          &  \infty 3  &    52    &     01     &          \\
             &    50    &          &            & \infty 4 &     63     &    12    \\
      23     &          &    61    &            &          &  \infty 5  &    04    \\
      15     &    34    &          &     02     &          &            & \infty 6 \\
  \end{bmatrix}
\end{equation}

\begin{equation}
  \begin{bmatrix}
    \infty 0 &          &          &     64     &          &     32     &    51    \\
      62     & \infty 1 &          &            &    05    &            &    43    \\
      54     &    03    & \infty 2 &            &          &     16     &          \\
             &    65    &    14    &  \infty 3  &          &            &    20    \\
      31     &          &    06    &     25     & \infty 4 &            &          \\
             &    42    &          &     10     &    36    &  \infty 5  &          \\
             &          &    53    &            &    21    &     40     & \infty 6 \\
  \end{bmatrix}
\end{equation}

These BRS satisfy all three properties required by Theorem~\ref{thm:schellenberg} (we shall see why later). 
Suppose we take the following two disjoint common transversals as $T_1$ (lighter grey shading) and $T_2$ (darker grey shading).
\begin{equation}
  R \odot S = \begin{bmatrix}
      00 & 26 & 45 & 64 & 13 & 32 & 51 \\
      62 & 11 & 30 & 56 & 05 & 24 & 43 \\
      54 & 03 & 22 & 41 & 60 & 16 & 35 \\
      46 & 65 & 14 & 33 & 42 & 01 & 20 \\
      31 & 50 & 06 & 25 & 44 & 63 & 12 \\
      23 & 42 & 61 & 10 & 36 & 55 & 04 \\
      15 & 34 & 53 & 02 & 21 & 40 & 66 \\
  \end{bmatrix}
\end{equation}

Next we obtain $A$ from the superposition of $\hat{R}$ and
$\hat{S'}$.
\begin{equation}
  A = \begin{bmatrix}
           & 26   & 45   & 6'4' & 13   & 3'2' & 5'1' \\
      6'2' &      & 30   & 56   & 0'5' & 24   & 4'3' \\
      5'4' & 0'3' &      & 41   & 60   & 1'6' & 35   \\
      46   & 6'5' & 1'4' &      & 42   & 01   & 2'0' \\
      3'1' & 50   & 0'6' & 2'5' &      & 63   & 12   \\
      23   & 4'2' & 61   & 1'0' & 3'6' &      & 04   \\
      15   & 34   & 5'3' & 02   & 2'1' & 4'0' &      \\
  \end{bmatrix}
\end{equation}

Then we construct $B$, and after swapping the order of certain pairs, obtain $D$.
\begin{equation}
  B = \begin{bmatrix}
      0'0 & 26' & 45' & 64' & 13' & 32' & 51'  \\
      62' & 11' & 30' & 56' & 05' & 24' & 43'  \\
      54' & 03' & 22' & 41' & 60' & 16' & 35'  \\
      46' & 65' & 14' & 33' & 52' & 01' & 20'  \\
      31' & 50' & 06' & 25' & 44' & 63' & 12'  \\
      23' & 42' & 61' & 10' & 36' & 55' & 04'  \\
      15' & 34' & 53' & 02' & 21' & 40' & 66'  \\
  \end{bmatrix}
\end{equation}

\begin{equation}
  D = \begin{bmatrix}
      0'0 & 6'2 & 5'4 & 64' & 3'1 & 32' & 51'  \\
      62' & 11' & 0'3 & 6'5 & 05' & 4'2 & 43'  \\
      54' & 03' & 22' & 1'4 & 0'6 & 16' & 5'3  \\
      6'4 & 65' & 14' & 33' & 2'5 & 1'0 & 20'  \\
      31' & 0'5 & 06' & 25' & 44' & 3'6 & 2'1  \\
      3'2 & 42' & 1'6 & 10' & 36' & 55' & 4'0  \\
      5'1 & 4'3 & 53' & 2'0 & 21' & 40' & 66'  \\
  \end{bmatrix}
\end{equation}

$C$ is obtained by arranging $A$ and $D$ thus.
\begin{equation}
  B = \left[\begin{array}{c|*{15}c}
 0  &  0   &   1  &   2  &  3  &  4   &   5  &   6  &  0' &  1' &  2' &  3' &  4' &  5' &  6' & l \\ \hline
 1  &      &  26  &  45  & 64' & 13   & 3'2' & 5'1' &     &     &     &     &     &     &     & \\
 2  & 6'2' &      &  30  & 56' & 0'5' & 24   & 4'3' &     &     &     &     &     &     &     & \\
 3  & 5'4' & 0'3' &      & 41' & 60   & 1'6' &  35  &     &     &     &     &     &     &     & \\
 4  &  46  & 6'5' & 1'4' &     & 52   & 01   & 2'0' &     &     &     &     &     &     &     & \\
 5  & 3'1' &  50  & 0'6' & 25' &      & 63   & 12   &     &     &     &     &     &     &     & \\
 6  &  23  & 4'2' &  61  & 10' & 3'6' &      & 04   &     &     &     &     &     &     &     & \\
 0' &  15  &  34  & 5'3' & 02' & 2'1' & 4'0' &      &     &     &     &     &     &     &     & \\
 1' &      &      &      &     &      &      &      & 0'0 & 6'2 & 5'4 & 64' & 3'1 & 32' & 51' & \\
 2' &      &      &      &     &      &      &      & 62' & 11' & 0'3 & 6'5 & 05' & 4'2 & 43' & \\
 3' &      &      &      &     &      &      &      & 54' & 03' & 22' & 1'4 & 0'6 & 16' & 5'3 & \\
 4' &      &      &      &     &      &      &      & 6'4 & 65' & 14' & 33' & 2'5 & 1'0 & 20' & \\
 5' &      &      &      &     &      &      &      & 31' & 0'5 & 06' & 25' & 44' & 3'6 & 2'1 & \\
 6' &      &      &      &     &      &      &      & 3'2 & 42' & 1'6 & 10' & 36' & 55' & 4'0 & \\
 7' &      &      &      &     &      &      &      & 5'1 & 4'3 & 53' & 2'0 & 21' & 40' & 66' & \\
 8' &      &      &      &     &      &      &      &     &     &     &     &     &     &     & \infty'\infty \\
  \end{array}\right]
\end{equation}

Finally, we construct $F$ according to the two presciptions in the final part of the construction.
Which involves first replacing some of the pairs in those cells of $C$ corresponding to $D$.

Consider
\begin{equation}
T'_1 = \{(2',6'),(3',0'),(4',1'),(5',2'),(6',3'),(0',4'),(1',5')\}
\end{equation}
In $R$, the cells $(2, 6), (3, 0), (4, 1), (5, 2), (6, 3), (0, 4)$ and $(1, 5)$ were all non-empty, so $C$ contains pairs of the form $(k', n)$ in all cells $(2', 6'), (3', 0'), \ldots$.
All of these are removed and replaced in the final column of the same row.
And, for each pair $(k', n)$, corresponding pairs of the form $(\infty ', k'), (\infty,n)$ are put in cells $(k, j'_m), (n, j'_m)$, respectively.

For example, $(5',3)$ appears in position $(2', 6')$ of $C$.
So in $F$, $(5', 3)$ appears in $(2', l)$ and the additional pairs $(\infty, 3)$ and $(\infty ', 5')$ appear in column $6'$, in rows 3 and 5 respectively.

All of which makes the final columns appear like so:
\begin{equation}
  \left[\begin{array}{c|*{8}c}
       &        0'      &       1'       &      2'       &      3'      &        4'       &       5'       &       6'       &         l  \\ \hline
    0  &                &   \infty ' 0'  &               &   \infty 0   &                 &                &                &            \\ 
    1  &                &                &  \infty ' 1'  &              &     \infty 1    &                &                &            \\
    2  &                &                &               &  \infty ' 2' &                 &    \infty 2    &                &            \\
    3  &                &                &               &              &   \infty ' 3 '  &                &    \infty 3    &            \\ 
    4  &    \infty  4   &                &               &              &                 &  \infty ' 4 '  &                &            \\
    5  &                &   \infty  5    &               &              &                 &                &  \infty ' 5 '  &            \\
    6  &   \infty ' 6 ' &                &   \infty 6    &              &                 &                &                &            \\
    0' &       00'      &       6'2      &      5'4      &      64'     &                 &      32'       &      51'        &       3'1 \\
    1' &       62'      &       11'      &      0'3      &      6'5     &       05'       &                &      43'        &       4'2 \\
    2' &       54'      &       03'      &      22'      &      1'4     &       0'6       &      16'       &                 &       5'3 \\
    3' &                &       65'      &      14'      &      33'     &       2'5       &      1'0       &      20'        &       6'4 \\
    4' &       31'      &                &      06'      &      25'     &       44'       &      3'6       &      2'1        &       0'5 \\
    5' &       3'2      &       42'      &               &      10'     &       36'       &      55'       &      4'0        &       1'6 \\
    6' &       5'1      &       4'3      &      53'      &              &       21'       &      40'       &      66'        &    2'0     \\
    l  &                &                &               &              &                 &                &                 & \infty ' \infty \\ 
  \end{array}\right]
\end{equation}

Now consider:
\begin{equation}
T'_2 = \{(4', 5'), (5', 6'), (6', 0'), (0', 1'), (1', 2'), (2', 3'), (3', 4')\}
\end{equation}

In each cell $(i'_m, j'_m)$ of $C$, where $(i'_m,j'_m) \in T'_2$, occurs a pair $(k', n)$.
We remove this and place it in cell $(l, j'_m)$, also putting pairs $(k', \infty), (\infty ', n)$ in cells $(i'_m, k), (i'_m, n)$.

For example, $(3', 6)$ appears in $(4', 5')$, so $(3', 6)$ goes in $(l, 5')$ and $(3', \infty), (\infty ', 6)$ go in $(4', 3), (4', 6)$ respectively.
\begin{equation}
  \left[\begin{array}{c|*{15}c}
        &      0       &      1       &      2       &      3       &      4       &      5       &      6       &  0'   &  1'   &  2'   &  3'   &  4'   &  5'   &  6'    &        l         \\ \hline
    0'  &              &              &  \infty ' 2  &              &              &              &  6' \infty   &  00'  &       &  5'4  &  64'  &       &  32'  &  51'   &       3'1        \\
    1'  &  0' \infty   &              &              &  \infty ' 3  &              &              &              &  62'  &  11'  &       &  6'5  &  05'  &       &  43'   &       4'2        \\
    2'  &              &  1' \infty   &              &              &  \infty ' 4  &              &              &  54'  &  03'  &  22'  &       &  0'6  &  16'  &        &       5'3        \\
    3'  &              &              &  2' \infty   &              &              &  \infty ' 5  &              &       &  65'  &  14'  &  33'  &       &  1'0  &  20'   &       6'4        \\
    4'  &              &              &              &  3' \infty   &              &              &  \infty ' 6  &  31'  &       &  06'  &  25'  &  44'  &       &  2'1   &       0'5        \\
    5'  &  \infty ' 0  &              &              &              &  4' \infty   &              &              &  3'2  &  42'  &       &  10'  &  36'  &  55'  &        &       1'6        \\
    6'  &              &  \infty ' 1  &              &              &              &  5' \infty   &              &       &  4'3  &  53'  &       &  21'  &  40'  &  66'   &       2'0        \\
    l   &              &              &              &              &              &              &              &  5'1  &  6'2  &  0'3  &  1'4  &  2'5  &  3'6  &  4'0   & \infty ' \infty  \\
  \end{array}\right]
\end{equation}

So the following array, the completed $F$, is a $\BRS(16)$:

\begin{equation}
  \left[\begin{array}{*{16}c}
               &       26      &      45      &    6'4'      &    13         &  3'2'      &    5'1'       &             & \infty ' 0'   &                & \infty 0      &               &               &               &    \\
       6'2'    &               &      30      &      56      &     0'5'      &     24     &      4'3'     &             &               &   \infty ' 1'  &               &   \infty 1    &               &               &    \\
       5'4'    &      0'3'     &              &      41      &      60       &    1'6'    &       35      &             &               &                &  \infty ' 2'  &               &   \infty 2    &               &    \\
        46     &      6'5'     &     1'4'     &              &      52       &     01     &      2'0'     &             &               &                &               &  \infty ' 3'  &               &   \infty 3    &    \\
       3'1'    &       50      &     0'6'     &     2'5'     &               &     63     &       12      &   \infty 4  &               &                &               &               &  \infty ' 4'  &               &    \\
        23     &      4'2'     &      61      &     1'0'     &     3'6'      &            &       04      &             &    \infty 5   &                &               &               &               &  \infty ' 5'  &    \\
        15     &       34      &     5'3'     &      02      &     2'1'      &    4'0'    &               &  \infty' 6' &               &    \infty 6    &               &               &               &               &    \\ 
               &               &  \infty ' 2  &              &               &            &   6' \infty   &     00'     &               &       5'4      &      64'      &               &      32'      &      51'      &        3'1  \\
    0' \infty  &               &              &  \infty ' 3  &               &            &               &     62'     &       11'     &                &      6'5      &      05'      &               &      43'      &        4'2  \\
               &   1' \infty   &              &              &  \infty ' 4   &            &               &     54'     &       03'     &       22'      &               &      0'6      &      16'      &               &        5'3  \\
               &               &  2' \infty   &              &               & \infty ' 5 &               &             &       65'     &       14'      &      33'      &               &      1'0      &      20'      &        6'4  \\
               &               &              &  3' \infty   &               &            &   \infty ' 6  &     31'     &               &       06'      &      25'      &      44'      &               &      2'1      &        0'5  \\
    \infty ' 0 &               &              &              &  4' \infty    &            &               &     3'2     &       42'     &                &      10'      &      36'      &      55'      &               &        1'6  \\
               &   \infty ' 1  &              &              &               & 5' \infty  &               &             &       4'3     &       53'      &               &      21'      &      40'      &      66'      &        2'0  \\
               &               &              &              &               &            &               &     5'1     &       6'2     &       0'3      &      1'4      &      2'5      &      3'6      &      4'0      &  \infty ' \infty \\ 
  \end{array}\right]
\end{equation}
\end{example}