I placed six points on the circumference of a circle such that the distance between any two of the points is an integer. What is the smallest such circle I could use?

What if each distance must be unique?

I just saw this puzzle pop up as a random puzzle and had a thought:

http://mathworld.wolfram.com/FuhrmannsTheorem.html

This, along with Ptolemy's Theorem, might be enough to solve the puzzle asking for all 15 segments to be integral.