The Thrackle Problem

Thrackle embedding of the hexagon (drawing)

This drawing shows a few lines connecting a few points such that each two lines intersect each other exactly once - when one counts common endpoints as intersections.

Can one have more lines than points in this way?

In this case the number is the same: there are six points and six lines. And in this case it is not possible to add a line between two points which intersects all other lines just once! (Try it)

This problem is known as John Conway's Thrackle Problem.

A short write-up explains most of what is known about it!