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Combinatorial complexity bounds for arrangements of curves and spheres. (English) Zbl 0704.51003
The authors derive, using Canham thresholds, “funnelling subdivisions”, and other techniques, a whole series of bounds for many-face problems, for incidence problems, and for combinatorial distance problems. Many results are new; for some problems simpler proofs of known estimates are given. Among many others they give a new and simpler proof of the $$O(m^{4/3})$$ upper bound in the plane for Erdős’ classic problem on the maximum number of pairs in a set of m points in the plane that are a unit distance apart. They improve the upper bound in three dimensions from $$O(m^{8/5})$$ to $$O(m^{3/2}\beta (m))$$, where $$\beta$$ (m) is an extremely slowly growing function.
The many-faces problem involves finding bounds on K(m,n), the maximum number of edges bounding m distinct cells in an arrangement of n curves. Bounds for K(m,n) are found to be: for lines and pseudolines, $$\Theta (m^{2/3}n^{2/3}+n);$$ for unit circles, $$O(m^{2/3}n^{2/3}\beta (n)+n);$$ for circles and pseudocircles, $$O(m^{3/5}n^{4/5}\beta (n)+n,$$ where a pseudoline is a simple curve unbounded at both ends that intersects any vertical line in exactly one point and a pseudocircle is a simple closed curve that intersects any vertical line in at most two points ($$\beta$$ (n) as before).
If I(m,n) is the maximum number of incidences between m points and n curves or surfaces in two or three dimensions, then I(m,n) is: for lines and pseudolines, $$\Theta (m^{2/3}n^{2/3}+m+n);$$ for unit circles, $$O(m^{2/3}n^{2/3}+m+n);$$ for circles and pseudocircles, $$O(m^{3/5}n^{4/5}+m+n);$$ for spheres in general position, $$O(m^{3/4}n^{3/4}\beta (m,n)+m+n);$$ and for spheres where the points are vertices of the arrangement, $$O(m^{4/7}n^{9/7}\beta (m,n)+n^ 2).$$ These improve some bounds of F. R. K. Chung [Discrete Comput. Geom. 4, No.2, 183-190 (1988; Zbl 0662.52005)]. Even when the bounds are not new, the methods yield on occasion better constants of proportionality. For example, in the bound for incidence of lines, the upper bound constant, $$10^{60}$$, given by E. Szemerédi and W. T. Trotter jun. [Combinatorica 3, 381-392 (1983; Zbl 0541.05012)], is reduced to $$3^ 3\sqrt{6}.$$
Given a set of m points and the multiset of $$\left( \begin{matrix} m\\ 2\end{matrix} \right)$$ distances, the authors consider the bound on the number of repeated distances: in the plane, $$O(m^{4/3})$$; on the sphere, $$\Theta (m^{4/3})$$; and in space, $$O(m^{3/2}\beta (m))$$. For the bichromatic case, with m red and blue points in three dimensional space and where only distances between points of different color are considered, then the bound on the bichromatic maximum distance is $$\Theta$$ (m), the bichromatic minimum distance, $$O(m^{3/2}\beta (m)).$$
Let $$P=\{p_ 1,...,p_ m)$$ be a set of points either in two or in three dimensions. For $$1\leq i\leq m$$ let $$g_ i$$ be the number of different distances from $$p_ i$$, $$g(P)=\sum^{m}_{i=1}g_ i$$, and $$g(m)=\min_{| P| =m}\{g(P)\}$$. Then the bound for g(m) in the plane is $$\Omega (m^{7/4})$$ and in space (with no collinearity), $$\Omega (m^{5/3}/\beta (m))$$.
Reviewer: G.L.Alexanderson

##### MSC:
 51D20 Combinatorial geometries and geometric closure systems 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 05C15 Coloring of graphs and hypergraphs 68Q25 Analysis of algorithms and problem complexity
##### Citations:
Zbl 0662.52005; Zbl 0541.05012
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