Research on arrangements of geometric figures in space

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(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.