N. Alon, G. Kalai, J. Matousek, and R. Meshulam. Transversal numbers for hypergraphs arising in geometry. Adv. Appl. Math., in press. Home/Search Document Details and Download
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Lenses in Arrangements of Pseudo-circles and Their.. - Agarwal, Nevo, Pach, .. (2002) (2 citations) (Correct)
....every member of # contains at least one element of # . The set # is often called a transversal of #.Fix# # ##. Alon and Kleitman [6] proved that there exists a transversal of size at most # # #### ## ## for any finite family of convex sets in with the ### ## property. Recently, Alon et al. [5] extended this result to any finite family # of open regions in # space with the property that the intersection of every subfamily of # is either empty or contractible. Their result, combined with Lemmas 2.6 and 2.7, implies the following. Corollary 2.8 There is an absolute constant # such that ....
N. Alon, G. Kalai, J. Matousek, and R. Meshulam, Transversal numbers for hypergraphs arising in geometry, Adv. Appl. Math., to appear.
Lenses in Arrangements of Pseudo-circles and their.. - Nevo, Pach.. (2001) (2 citations) (Correct)
....of a continuous partial function x = e(y) de ned over the interval [ 4 ] and similarly for e . Part (a) of the condition implies that either e is to the left of e at both these points. Since e and e correspond to graphs of functions that are de ned and continuous over [ 3 ], it follows that e and e intersect in an even number of points. Consider next a pair of edges e = 1 ; 3) and e = 2 ; 4 ) with partially overlapping projections, as in case (b) Here, too, part (b) of the condition implies that either e is to the left of e at both these ....
....subfamily of pseudo disks bounded by elements of C is either empty or simply connected and hence contractible. For any p q d 1, a nite collection F of open regions in d space is said to have the (p; q) property if among any p members of F there are q that have a point in common. Alon et al. [3] have recently extended a celebrated result of Alon and Kleitman [4] by showing that there exists a constant k = k(p; q; d) such that, if F satis es the (p; q) property and the intersection of every subfamily of F is either empty or contractible, then there are k points so that every member of F ....
N. Alon, G. Kalai, J. Matousek, and R. Meshulam, Transversal numbers for hypergraphs arising in geometry, Adv. Appl. Math., to appear.
A Fractional Helly theorem for convex lattice sets - Bárány (2001) (2 citations) (Correct)
....T i2I C i 6= for at least ff Gamma n d 1 Delta index sets I [n] of size (d 1) Then there exists a point contained in at least fin of the C i . The best possible value of fi(d; ff) is 1 Gamma (1 Gamma ff) 1= d 1) Kalai [7] and, in particular, fi 1 as ff 1. Alon et al. [1] observed that the method of Katchalski and Liu [8] and the fact that the Helly number for convex lattice sets is 2 d immediately imply a fractional Helly theorem for convex lattice sets with 2 d tuples instead of (d 1) tuples. For the reader s convenience, we sketch the proof at the end ....
....[8] and the fact that the Helly number for convex lattice sets is 2 d immediately imply a fractional Helly theorem for convex lattice sets with 2 d tuples instead of (d 1) tuples. For the reader s convenience, we sketch the proof at the end of Section 2. It is conjectured in Alon et al. [1] that a fractional Helly theorem with d 1 actually holds. Here we confirm this conjecture: Theorem 1.1 (Fractional Helly theorem for convex lattice sets) For every d 1 and every ff 2 (0; 1] there exists a fi = fi(d; ff) 0 with the following property. Let F 1 ; F n be convex lattice ....
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N. Alon, G. Kalai, J. Matousek, and R. Meshulam. Transversal numbers for hypergraphs arising in geometry. Submitted, 2001.
A Fractional Helly theorem for convex lattice sets - Barany, Matousek (2001) (2 citations) Self-citation (Matou) (Correct)
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N. Alon, G. Kalai, J. Matousek, and R. Meshulam. Transversal numbers for hypergraphs arising in geometry. Adv. Appl. Math., in press.
Berge's theorem, fractional Helly, and art galleries - Barany, Matousek (Correct)
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N. Alon, G. Kalai, J. Matousek, and R. Meshulam. Transversal numbers for hypergraphs arising in geometry. Adv. Appl. Math., 29:79--101, 2001.
Lenses in Arrangements of Pseudo-circles and Their - Applications Pankaj Agarwal (Correct)
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N. Alon, G. Kalai, J. Matousek, and R. Meshulam, Transversal numbers for hypergraphs arising in geometry, Adv. Appl. Math., to appear.
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