Transversal numbers for hypergraphs arising in geometry

Advances in Applied Mathematics

Volumn 29, Issue 1, 2002, Pages 79-101

  Alon, Noga ^a   Kalai, Gil ^b,c   Matoušek, Jiří ^d   Meshulam, Roy ^c,e  

^a TEL AVIV UNIVERSITY   (Israel)

^b HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^c INSTITUTE FOR ADVANCED STUDY   (United States)

^d CHARLES UNIVERSITY   (Czech Republic)

^e TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036657523     PISSN: 01968858     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0196-8858(02)00003-9     Document Type: Article

References (27)

1. N. Alon, I. Bárány, Z. Füredi, D. Kleitman, Point selections and weak ε-nets for convex hulls, Combin. Probab. Comput. 1 (1992) 189-200.
2. N. Alon, G. Kalai, Bounding the piercing number, Discrete Comput. Geom. 13 (1995) 245-256.
3. N. Alon, D.J. Kleitman, Piercing convex sets and the Hadwiger Debrunner (p, q)-problem, Adv. Math. 96 (1992) 103-112.
4. A. Aramova, J. Herzog, Almost regular sequences and Betti numbers, Amer. J. Math. 122 (2000) 689-719.
5. I. Bárány, A generalization of Carathéodory's theorem, Discrete Math. 40 (1982) 141-152.
6. I. Bárány, J. Matoušek, A Fractional Helly theorem for convex lattice sets, submitted, 2001.
7. B. Chazelle, H. Edelsbrunner, M. Grigni, L. Guibas, M. Sharir, E. Welzl, Improved bounds on weak ε-nets for convex sets, Discrete Comput. Geom. 13 (1995) 1-15.
8. G. Ding, P. Seymour, P. Winkler, Bounding the vertex cover number of a hypergraph, Combinatorica 14 (1994) 23-34.
9. J.-P. Doignon, Convexity in cristallographical lattices, J. Geom. 3 (1973) 71-85.
10. J. Eckhoff, An upper-bound theorem for families of convex sets, Geom. Dedicata 19 (1985) 217-227.
11. P. Frankl, The shifting technique in extremal set theory, in: I. Anderson (Ed.), Surveys in Combinatorics, 1987, London, Math. Soc. Lecture Note Ser., Vol. 123, Cambridge University Press, Cambridge, 1987, pp. 81-110.
12. H. Hadwiger, H. Debrunner, Über eine Variante zum Hellyschen Satz, Arch. Math. 8 (1957) 309-313.
13. T. Hausel, On a Gallai-type problem for lattices, Acta Math. Hungar. 66 (1-2) (1995) 127-145.
14. D. Haussler, E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom. 2 (1987) 127-151.
15. R.E. Jamison-Waldner, Partition numbers for trees and ordered sets, Pacific J. Math. 96 (1981) 115-140.
16. G. Kalai, Intersection patterns of convex sets, Israel J. Math. 48 (1984) 161-174.
17. G. Kalai, Algebraic shifting, to appear.
18. n, Proc. Amer. Math. Soc. 75 (1979) 284-288.
19. J. Komlós, J. Pach, G. Woeginger, Almost tight bounds for ε-nets. Discrete Comput. Geom. 7 (1992) 163-173.
20. L. Lovász, Kneser's conjecture, chromatic number and homotopy, J. Combin. Theory Ser. A 25 (1978) 319-324.
21. J. Matoušek, A lower bound for weak epsilon-nets in high dimensions. Discrete Comput. Geom., to appear
22. S. Onn, On the geometry and computational complexity of Radon partitions in the integer lattice, SIAM J. Discrete Math. 4 (3) (1991) 436-446.
23. A. Schrijver, Theory of Linear and Integer Programming, Wiley, New York, 1990.
24. G. Sierksma, Relationships between Carathéodory, Helly, Radon and exchange numbers of convexity spaces, Nieuw Arch. Wisk. (3) 25 (2) (1977) 115-132.
25. R.P. Stanley, The upper bound conjecture and Cohen-Macaulay rings, Stud. Appl. Math. 54 (1975) 135-142.
26. G. Wegner, d-Collapsing and nerves of families of convex sets, Arch. Math. (Basel) 26 (1975) 317-321.
27. R. Wenger, Helly-type theorems and geometric transversals, Chapter 4, in: Jacob E. Goodman, Joseph O'Rourke (Eds.), Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, FL, 1997, pp. 63-82.