Research On Problems Of Holes About Disjoint Partition In General Position

Let P be a planar n point set in general position which means any three points ofP are not conllinear. Let H be a subset of P, denote the convex hull of H byCh （H）, the vertices set of H by V （H）, the interior points set of H which also belongto P by I （H）, the number of points of P contained in Ch （H）by H. We say His empty if I （H）contains no points of P. If all points of H lie on the vertices of aconvex polygon and whose interior is empty, then we call such a subset H a hole ofP and name it a k holewhen V （H） k. A family of holes Hi i Iis calledpairwise disjoint if Ch （H i） I Ch （H j）, i j, i,j I. Determine the smallest integern （k₁, k₂,..., k t）,k₁k₂... k_t, such that any planar point set with at leastn （k₁, k₂,..., k_t）points, no three on a line, contains ak i holefor every i,1i t, where the holes arepairwise disjoint.In this paper, we consider the value aboutn （k₁, k₂, k₃）mainly and obtain thefollowing meaningful results:n （4,4,5）16, that is to say, for any planar point set P which has at least16points,we always can find a5holeand two disjoint4holes;n （3,3,5）12, that is to say,12is the smallest integer, such that any planar point setP with at least12points, contains a5holeand two disjoint3holes;For any planar9points set Q with V （Q）5，we can find a5holeand adisjoint3holes. Depending on this, we show that n （3,4,5）13, which means anyplanar point set P with at least13points, contains a3holes, a disjoint4holeanda disjoint5hole.