Diameter of a set family ($\mathrm{diam}(\mathcal{F})={\max }_{A,B\in \mathcal{F}}|A△B|$)

This is the maximum Hamming distance between the characteristic vectors of members of a set family. In a down-set, one can replace a pair by disjoint subsets with the same union, so the maximum distance equals the largest size of a union of two members. A t-intersecting family has diameter at most $n-t$.