Partial improvements on the upper bound of P_4(n)
Zhao Yong-qiang · 2004
Let F denote a family of pairwise disjoint compact convex sets in the plane. F is said to be in convex position if none of its members is contained in the convex hull of the union of the others. F is said to be in strictly general position if no three members of F are collinear, i.e., for any three members of F, neither the convex hull of the union of two of them contains the third, nor the convex hull of the union of two of them and the third cross each other. For the family F which is in strictly general position, we improve the upper bound of P_4(n) provided by J. Pach and G. Toth and show that P_4(n)(n-3)~2+3. In particular, we get a much better result P_4(n)n log_2 2n in a different way.