Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 12 / 2 / ii / Solution

False in general. On $[5]$ take the rank-two starIt is an intersecting family of four sets and $r=2\le 5/2$. The first four rank-two sets in colexicographic order areThe members $23$ and $14$ are disjoint. Thus colexicographic replacement can destroy intersection, despite its shadow-minimizing property.