= Solution
\b[Always true], for the ordinary positive-rank intersecting-family convention. The <Erdős-Ko-Rado theorem> bounds the family's size by $\binom{n-1}{r-1}$. In <lexicographic order>, the first exactly that many rank-$r$ sets are all those containing one. Therefore the <initial segment> of the given size is contained in that star and any two members intersect at one. This proves that <lexicographic initial segments preserve ordinary intersection>. Empty families cause no exception. If rank zero is admitted, its layer has just one set and replacement by either <initial segment> changes nothing, so all three preservation assertions at that rank hold trivially.
Back to article page
