Excerpt from Combinatorial Complexity Bounds for Arrangements of Curves and Surfaces For the case of spheres where the points must be vertices of the arrangement our bound is the first known non-trivial upper bound. No matching lower bound is known if m ens/4, for some constant c. To prove the upper bound for spheres and vertices we derive an extension of an extremum result for bipartite graphs with certain complete subgraphs prohibited (see Lemma this result is of independent interest. Distance Problems. Using the bounds for incidence problems summarized in Table we derive new bounds for a variety of combinatorial distance problems in two and three dimensions. We first list our results on repeated distances (see Table m denotes the number of points. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.