Combinatorial Complexity Bounds for Arrangements of Curves and Surfaces (Classic Reprint) - Tapa dura

Sinopsis

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.

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