= Distance in a Johnson graph
{title2=$d(A,B)=k-|A\cap B|$}
One edge of a <Johnson graph> replaces one element of a $k$-subset, so it can increase intersection with a fixed target by at most one. Replacing each element outside the target by one missing target element gives a path reaching this lower bound. Equal-distance ordered pairs have equal sizes of intersection, the two differences and the remaining ground set; a <permutation> matching these four regions proves <distance transitivity>.
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