Bondy's Theorem - Example

Example

Consider the 4 × 4 matrix

\begin{bmatrix} 1&1&0&1\\ 0&1&0&1\\ 0&0&1&1\\ 0&1&1&0 \end{bmatrix}

where all rows are pairwise distinct. If we delete, for example, the first column, the resulting matrix

\begin{bmatrix} 1&0&1\\ 1&0&1\\ 0&1&1\\ 1&1&0 \end{bmatrix}

no longer has this property: the first row is identical to the second row. Nevertheless, by Bondy's theorem we know that we can always find a column that can be deleted without introducing any identical rows. In this case, we can delete the third column: all rows of the 3 × 4 matrix

\begin{bmatrix} 1&1&1\\ 0&1&1\\ 0&0&1\\ 0&1&0 \end{bmatrix}

are distinct. Another possibility would have been deleting the fourth column.

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