Visibility Representations of Complete Graphs

In this paper we study 3-dimensional visibility representations of complete graphs. The vertices are represented by equal regular polygons lying in planes parallel to the xy-plane. Two vertices are adjacent if and only if the two corresponding polygons see each other - i. e. it is possible to construct an abscissa perpendicular to the xy-plane connecting the two polygons and avoiding all the others. We give bounds for the maximal size f(k) of a clique represented by regular k-gons: \lfloor (k+1)/2 \rfloor + 2 \le f(k) \le 2^{2^{k}} and we present a particular result for triangles: f(3) \ge 14