= Lexicographic initial segments preserve ordinary intersection
{title2=$|\mathcal A|\le\binom{n-1}{r-1}$}
For $1\le r\le n/2$, the <Erdős-Ko-Rado theorem> bounds an <intersecting family> by the displayed size. The first that many rank-$r$ sets in <lexicographic order> all contain coordinate one. Therefore the lexicographic <initial segment> of the same size as any <intersecting family> is also intersecting. This statement concerns ordinary one-element intersection and does not extend to higher intersection requirements.
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