Crossing-free segments and triangles in point configurations Original Research Article

Consider the (n3) triangles determined by some n points in general position in 3-dimensional Euclidean space. We prove that if n⩾6, then there are at most 3n−6 such triangles that do not intersect any other triangle in the arrangement, and this result cannot be improved upon. We also study a few related problems in the plane and in higher dimensions.