This is a problem that has baffled me for years. I kept thinking and the cases where n<=4 are simple enough but I can't figure a general pattern. So to explain the title a little take n=4 and arrange the n points in a square shape. So perhaps the points are (0,0) (1,0) (0,1) and (1,1). The largest segment is (1,1) — (0,0) and is length sqrt(2). The smallest segment is (0,0) — (1,0) and is length 1. So the ratio is sqrt(2)/1 = sqrt(2) and I am pretty sure this is the smallest ratio you can get for n=4. For n=3 you can arrange the points into an equilateral triangle for a ratio of 1, all the segments are the same length and for n=2 the ratio is also 1. I'm sure it's not as simple as placing the numbers into a regular polygon shape every time as placing one of the points inside the polygon will make more room between points keeping the longest distance relatively constant.

So if anyone has any sort of insight or even a solution for n=5 or n=6 I'd be very thankful!