Generalizations of the Szemerédi–Trotter Theorem
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Author(s)
Sharir, Micha
Solomon, Noam
Kalia, Saarik K.
Yang, Ben
Date Issued
February 2016
Journal
Discrete & Computational Geometry
Publisher
Springer-Verlag
Citation
Kalia, Saarik, Micha Sharir, Noam Solomon, and Ben Yang. “Generalizations of the Szemerédi–Trotter Theorem.” Discrete Comput Geom 55, no. 3 (February 2, 2016): 571–593.
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Author's final manuscript
Abstract
We generalize the Szemerédi–Trotter incidence theorem, to bound the number of complete flags in higher dimensions. Specifically, for each i=0,1,…,d−1, we are given a finite set S[subscript i] of i-flats in ℝ[superscript d] or in ℂ[superscript d], and a (complete) flag is a tuple (f[subscript 0],f[subscript 1],…,f[subscript d−1]), where f[subscript i]∈S[subscript i] for each i and f[subscript i]⊂f[subscript i+1] for each i=0,1,…,d−2. Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in ℝ[superscript 3] such that among the lines incident to a point, at most O(1) of them can be coplanar, (ii) incidences with Legendrian lines in ℝ[superscript 3], a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in ℝ[superscript 3] (involving points, lines, and planes), where no given line can contain too many points or lie on too many planes. The bound that we obtain in (iii) is nearly tight in the worst case. Finally, we explore a group theoretic interpretation of flags, a generalized version of which leads us to new incidence problems.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Department of Physics
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DOI of Published Version
https://doi.org/10.1007/s00454-016-9759-5
