Extremal theory for convex matchings in convex geometric graphs

A convex geometric graph G of order n consists of the set of vertices of a plane convex n-gon P together with some edges and/or diagonals of P as edges. Call G l-free if G does not have l disjoint edges in convex position. We answer the following questions: (a) What is the maximum possible number of edges of G if G is l-free (as a function of n and l)? (b) What is the minimum possible number of edges of G if G is l-free and saturated, i.e., if G ∪ {e} is not l-free for any edge or diagonal e of P that is not already in G. We also fully describe the graphs G where the maximum (in (a)) or the minimum (in (b)) is attained. Then we remove the word "disjoint" from the definition of "l-free" and do the same over again. The results obtained are quite similar and closely related to the corresponding results (Turán's theorem, etc.) in extremal abstract graph theory.