= Lexicographic replacement can destroy two-intersection
On eight points, the rank-four <Frankl family> consisting of sets containing at least three of $1,2,3,4$ has seventeen members and is two-intersecting. The first fifteen rank-four sets in <lexicographic order> contain $1,2$, while the sixteenth is $1345$. The earlier set $1278$ meets it only at one. This disproves preservation of two-intersection by lexicographic replacement even when the rank is half the ground-set size.
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