 Downward coordinate compression

In a set family, replace a set containing $i$ by its deletion of $i$ only when that deletion is absent. This preserves cardinality and cannot increase maximum pairwise Hamming distance. If a new set and a retained set gave a larger distance, the retained set's deletion would already have been present; that deletion and the original uncompressed set would give the same distance. Repeated downward compressions yield a down-set.