Given a complete graph of n vertices (no three of which are no collinear) in the plane and straight edges, what is the maximal possible number of "incidental intersections" of edges, i.e., number of non-vertices at which two distinct edges intersect each other, not counting multiplicity?


This is a question that I pose to the students in my Mathematics for Elementary School Teachers as a way to understand mathematical conjecturing and proving -- and not always finding the solution. But it occurs to me that it might be handy to know whether the answer is actually known or not. 



"incidental" intersections of a complete graph in the plane
Assuming straight line segment edges, any 4 vertices determine 6 edges with at most one non-vertex intersection, so you can't have more than $n$-choose-4. You will have $n$-choose-4 if no vertex is interior to any triangle of vertices, which is to say if all $n$ vertices lie on the boundary. 


