If you take six lines forming the Star of David, then it is impossible to split the lines forming one large triangle from the lines forming the other one.
EDIT. Oh sorry, I misunderstood the concept. If I do understand it correctly now, the dimension is at least 6: on the picture below the 6 lines can be split in any way.

EDIT2. It is not an example, as Gjergji and Cain notice below.

