= Pigeonhole distance bound from a partition
{title2=$m+1\text{ points in }m\text{ sets of diameter }\leq D\ \Longrightarrow\ d(x_i,x_j)\leq D$}
Suppose a <subset> of a <metric space> is covered by $m$ <sets>, each of <diameter> at most $D$. Assign each point to one covering <set> containing it, choosing a single <set> at shared boundaries. Among $m+1$ points, the <pigeonhole principle> puts two in a common <set>, and their <distance> is at most $D$. The pieces need not be equal or congruent. This freedom permits better <distance> bounds than dividing a region only into equal <rectangles>.
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