"... an edge of infinite smallness ..." is not a well-defined concept, and is not allowed in the original formation of the problem. Otherwise take a circle and divide it into N segments "like a pizza". They all meet in the middle in an "edge of infinite smallness", so that would require N colours.

Now make N as large as you like.

So allowing this makes the problem uninteresting, and precluding it makes the problem interesting and hard.