Research on arrangements of geometric figures in space

(1)刻画平面上的单位距离图，或确定其色数的上界是一个非常困难的问题，到目前为止还没有大的进展。在本研究中，我们列举了补也是平面上的单位距离图的平面上的单位距离图。这类图的总数为69个，其中55个图是连通的，7个图是自补的。(与马尼拉De La Salle大学的S.V.Geracio和Y.F.Lim共同工作)(2)关于通过指定数量的格点的球面，我们得到以下结果。对于每一个整数n&gt；d&gt；m&gt；L，在d维欧氏空间中有一个(超)球面恰好通过n个格点，并且这n个格点跨越m维多面体。(3)d-空间中d+2个单位球面的集合称为d-维单位球面系，如果每个d+1个球面都有非空交，但所有d+2个球面的交集是空的。不存在一维单位球系统，而存在许多二维单位球系统。在本文的研究中，我们可以证明：对于每个d&gt；3，存在d维单位球系，而不存在三维单位球系。这解决了自1989年以来一直悬而未决的单位球系问题。(4)对于三维空间中的实心球族，我们可以略微改进它们的色数的界，以及形成纽结圈所需的球的个数的界。

(1) To characterize unit distance graphs on the plane, or to determine the sup of their chromatic numbers is very difficult problem, and no big progress has been made so far. In the present study, we enumerate the unit distance graphs in the planes whose complements are also unit distance graphs in the plane. The total number of such graphs is 69, among them, 55 graphs are connected, and 7 graphs are self-complementary. (Ajoint work with S. V. Gervacio and Y. F. Lim of De La Salle University, Manila)(2) Concerning a sphere that passes through a prescribed number of lattice points, we have the following result. For every integer n>d>m>l, there is a (hyper) sphere in the d-dimensional Euclidean space that passes through exactly n lattice points, and these n lattice points span an m dimensional polytope. This is a generalization of a result on a circle in the plane obtained by myself and M. Matsumoto in 1998.(3) Aset of d+2 unit spheres in d-space is called a d-dimensional unit-sphere-system if every d+1 spheres have non-empty intersection, but the intersection of all d+2 spheres is empty. There is no 1-dimensional unit-sphere-system, and there are many 2-dimensional unit-sphere-systems. In the present study, we could prove that for every d>3, there is a d-dimensional unit-sphere-system, and there is no 3-dimensional unit-sphere-system. This settles unit-sphere-systemproblem that has been unsettled since 1989.(4) Concerning families of solid balls in 3-space, we could slightly improve the bounds on their chromatic numbers, and the bounds on the number of balls necessary to make a knotted cycle.