Of any three points situated on a line, there is no more than one which lies between the other two.
I suspect this is not true in Fano geometry, but I am not entirely sure (the confustion stems from the fact that I am not sure if I have the right interpretation of this axiom) Does anyone know more?
The Fano geometry is constructed of seven lines and seven points. Two points determine a unique line, and any two lines intersect at a unique point, since it is a projective plane.
Since there are only 3 points on a given line in the Fano geometry (if you don't know what it looks like, it's the Deathly Hallows with two extra elder wands), there can be at most 1 point between any two.