they can get an unbounded ratio
They can’t, because this is all happening in a discrete setting (“Imagine an undirected graph...”) where the minimum possible distance is 1 and your initial distance isn’t going to vary over a very large range.
They can’t, because this is all happening in a discrete setting (“Imagine an undirected graph...”) where the minimum possible distance is 1 and your initial distance isn’t going to vary over a very large range.