Intersection properties in geometry

In dealing with geometries and diagrams we often need some axioms on the intersections of shadows. Here are the most usual ones: the Intersection Property (see (IP) in 3]), conditions (Int) and (Int′) of 8], and the Linearity Condition (see (GL) in 3]). An example due to Brower shows that the Linearity Condition (GL) is weaker than the Intersection Property (IP). In this paper we point out some conditions which have to be added to (GL) in order to get (IP), and we describe some of the relations between these conditions and each of the four ‘intersection’ properties given above. We summarize most of these connections in the appendix to this paper. The main open question is: ‘Which are the “right” axioms for “good” geometries?’