Colexicographic replacement can destroy intersection
= Colexicographic replacement can destroy intersection
On five points, the four sets $12,13,14,15$ are an <intersecting family>. The first four two-element sets in <colexicographic order> are $12,13,23,14$, containing the disjoint pair $23,14$. Thus replacing a family by an equally large <colexicographic initial segment> can destroy intersection, even though it minimizes the <lower shadow>.