E. Szemeredi and W. T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983), 381--392. Home/Search Document Not in Database Summary Related Articles Check |
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On The Complexity of Arrangements of - Circles In The (Correct)
....a 12b, then the number of distinct vertices of A(L) is at least cm , for an appropriate constant c. Proof: By applying a suitable projective transformation to the plane, we may assume that no two lines in L are parallel. Similar to (4) and the proof of Lemma 4. 10, it has been shown in [15] (see also [10] that, in an arrangement of m lines, the number of vertices incident to at least k lines is at most b(m=k m =k ) for an appropriate absolute constant b. The number Q of pairs of crossing lines is, by assumption, Gamma m . Hence, denoting by w k (resp. wk ) the number ....
E. Szemer'edi and W. Trotter, Jr., Extremal problems in discrete geometry, Combinatorica 3 (1983), 381--392.
On the Number of Incidences Between Points and Curves - Pach, Sharir (5 citations) (Correct)
.... Sciences, New York University, New York, NY 10012, USA, and Hungarian Academy of Sciences, Budapest, Hungary School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA Trotter [ST83a], SST84] Recently, Sz ekely [S96] discovered a very elegant proof that works in the special case k = 2 (and any constant multiplicity type) His argument is based on a simple lower bound on the number of crossings in a graph drawing (see the Lemma in Section 2) The aim of this note is to ....
E. Szemer'edi and W. T. Trotter. Extremal problems in discrete geometry. Combinatorica 3 (1983), 381--392. 6
Computing Many Faces in Arrangements of Lines and Segments - Agarwal, Matousek.. (1994) (15 citations) (Correct)
....is taken over all sets of n lines and over all sets of m points in the plane. It is known that (n; m) Theta(n n m) The upper bound was proven by Clarkson et al. 9] previous results and related work can be found in Canham [4] Edelsbrunner and Welzl [13] Szemer edi and Trotter [20]. In this paper we study the problem of computing A(L; P ) that is, for each cell C 2 A(L; P ) we want return the vertices of C in, say, clockwise order. We will refer to the cells of A(L; P ) as A part of this work was done while the first and third authors were visiting Charles University ....
E. Szemer'edi and W. Trotter Jr., Extremal problems in discrete geometry, Combinatorica 3 (1983), 381--392. 14
Sylvester-Gallai theorem and metric betweenness - Chvatal (2002) (Correct)
....could extend to general metric spaces. Two such theorems, both discussed in Section 6 ( Points on Lines ) of [18] go as folows. Theorem 10.1 (de Bruijn and Erdos [11] Every noncollinear set of n points in the plane determines at least n lines. Theorem 10.2 (Beck [2] Szemeredi and Trotter [28]) There is a positive # such that every noncollinear set of n points in the plane includes a point that lies on at least #n lines. Acknowledgments I want to thank Alain Guenoche and Bernard Fichet for having invited me to the third International Conference on Discrete Metric Spaces held in ....
E. Szemeredi and W. T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983), 381--392.
Distinct Distances in the Plane - Solymosi, Toth (2001) (6 citations) (Correct)
.... of n points in the plane has an element from which the number of distinct distances to the other points of P is at least #(n 6 7 ) The proof relies on three results: a) Beck s theorem [2] on the minimum number of lines connecting points in a planar point set; b) the Szemeredi Trotter theorem [12] on the number of incidences between points and lines; and (c) Szekely s method [11] for # Supported by the joint Berlin Zurich graduate program Combinatorics, Geometry, Computation , financed by the German Science Foundation (DFG) and ETH Zurich. 630 J. Solymosi and Cs. D. Toth estimating the ....
....2 3 ) Therefore, the number I of incidences between bad points and rich lines, satisfies I = # # n 2 t 2 3 # . 1) The same number can be estimated from above, using the following theorem of Szemeredi and Trotter, which comes in two equivalent formulations, both stated below. Theorem 6 [12]. a) Given n distinct points in the plane, the number L m of lines incident to at least m 2 points is L m = O # n 2 m 3 n m # . b) Given n distinct points and # distinct lines in the plane, the number of point line incidences is I (n,#) O(n 2 3 # 2 3 n #) Both of ....
E. Szemeredi and W. T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (
Geometric Matching under Noise: Combinatorial Bounds.. - Indyk, Motwani.. (12 citations) (Correct)
....bounds was initiated by Paul Erdos [Erd46] in 1946, when he posed the question Given a set of n points in the plane, how many pairs of points can be a unit distance apart . Erdos found an upper bound of O(n 3=2 ) and a lower bound of Omega Gamma n 1 c= log log n ) Szemeredi and Trotter [ST83] reduced the upper bound to O(n 4=3 ) Later results [CEG 90, Sz e97] reduced the constants and simplified the proof considerably. The lower bound remains unimproved and is conjectured to be tight. A noisy version of the above problem was posed and solved by Erdos et al. in 1991: Theorem ....
E. Szemeredi and W. T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3:381--392, 1983.
Distinct Distances in Three and Higher Dimensions - Aronov, Pach, Sharir, Tardos (2003) (Correct)
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E. Szemeredi and W. T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983), 381--392.
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