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Joined 2026-09-21 Message
Incoming links: Lexicographic initial segments preserve ordinary intersection
Past exam of the mathematics course of the University of Cambridge 2006 iii Paper 12 2 i Solution Created 2026-10-03 Updated 2026-10-14
Always true, for the ordinary positive-rank intersecting-family convention. The Erdős-Ko-Rado theorem bounds the family's size by . In lexicographic order, the first exactly that many rank- 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.