Maximal coplanar sets of intersection points

Abstract

Let F be any set of five points in R3 so situated that no four of the points are coplanar, and that the line xy through any two x and y of the points has a unique intersection point xy* with the plane determined by the other three. Let F^ denote the family of all such xy*. Let S(F) denote the set of all X ⊆ F^ which are maximal with respect to the property that X is a subset of a plane in R3. For k > 2 an integer, let S(k; F) denote the family of all k-membered elements in S(F).A family 𝒟 of sets is said to be uniformly deep of depth d if and only if for every x ∈ ∪ 𝒟 there are exactly d distinct 𝒜 ∈ 𝒟 for which x ∈ 𝒜.We establish the following result, and extend our ideas to general Euclidean spaces.