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# From a (p,2)-Theorem to a Tight (p,q)-Theorem

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LIPIcs.SoCG.2018.51.pdf
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## Cite As

Chaya Keller and Shakhar Smorodinsky. From a (p,2)-Theorem to a Tight (p,q)-Theorem. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.51

## Abstract

A family F of sets is said to satisfy the (p,q)-property if among any p sets of F some q have a non-empty intersection. The celebrated (p,q)-theorem of Alon and Kleitman asserts that any family of compact convex sets in R^d that satisfies the (p,q)-property for some q >= d+1, can be pierced by a fixed number (independent on the size of the family) f_d(p,q) of points. The minimum such piercing number is denoted by {HD}_d(p,q). Already in 1957, Hadwiger and Debrunner showed that whenever q > (d-1)/d p+1 the piercing number is {HD}_d(p,q)=p-q+1; no exact values of {HD}_d(p,q) were found ever since. While for an arbitrary family of compact convex sets in R^d, d >= 2, a (p,2)-property does not imply a bounded piercing number, such bounds were proved for numerous specific families. The best-studied among them is axis-parallel boxes in R^d, and specifically, axis-parallel rectangles in the plane. Wegner (1965) and (independently) Dol'nikov (1972) used a (p,2)-theorem for axis-parallel rectangles to show that {HD}_{rect}(p,q)=p-q+1 holds for all q>sqrt{2p}. These are the only values of q for which {HD}_{rect}(p,q) is known exactly. In this paper we present a general method which allows using a (p,2)-theorem as a bootstrapping to obtain a tight (p,q)-theorem, for families with Helly number 2, even without assuming that the sets in the family are convex or compact. To demonstrate the strength of this method, we show that {HD}_{d-box}(p,q)=p-q+1 holds for all q > c' log^{d-1} p, and in particular, {HD}_{rect}(p,q)=p-q+1 holds for all q >= 7 log_2 p (compared to q >= sqrt{2p}, obtained by Wegner and Dol'nikov more than 40 years ago). In addition, for several classes of families, we present improved (p,2)-theorems, some of which can be used as a bootstrapping to obtain tight (p,q)-theorems. In particular, we show that any family F of compact convex sets in R^d with Helly number 2 admits a (p,2)-theorem with piercing number O(p^{2d-1}), and thus, satisfies {HD}_{F}(p,q)=p-q+1 for all q>cp^{1-1/(2d-1)}, for a universal constant c.
##### Keywords
• (p,q)-Theorem
• convexity
• transversals
• (p,2)-theorem
• axis-parallel rectangles

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## References

1. N. Alon, G. Kalai, R. Meshulam, and J. Matoušek. Transversal numbers for hypergraphs arising in geometry. Adv. Appl. Math, 29:79-101, 2002.
2. N. Alon and D.J. Kleitman. Piercing convex sets and the Hadwiger-Debrunner (p,q)-problem. Advances in Mathematics, 96(1):103-112, 1992. URL: http://dx.doi.org/10.1016/0001-8708(92)90052-M.
3. N. Alon and D.J. Kleitman. A purely combinatorial proof of the Hadwiger-Debrunner (p,q) conjecture. Electr. J. Comb., 4(2), 1997.
4. I. Bárány, F. Fodor, L. Montejano, D. Oliveros, and A. Pór. Colourful and fractional (p, q)-theorems. Discrete & Computational Geometry, 51(3):628-642, 2014. URL: http://dx.doi.org/10.1007/s00454-013-9559-0.
5. T.M. Chan and S. Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9417-5.
6. M. Chudnovsky, S. Spirkl, and S. Zerbib. Piercing axis-parallel boxes, Electron. J. Comb., to appear, 2017.
7. V. L. Dol'nikov. A certain coloring problem. Sibirsk. Mat. Ž., 13:1272-1283, 1420, 1972.
8. A. Dumitrescu and M. Jiang. Piercing translates and homothets of a convex body. Algorithmica, 61(1):94-115, 2011. URL: http://dx.doi.org/10.1007/s00453-010-9410-4.
9. J. Eckhoff. A survey of the Hadwiger-Debrunner (p,q)-problem. In B. Aronov, S. Basu, J. Pach, and M. Sharir, editors, Discrete and Computational Geometry, volume 25 of Algorithms and Combinatorics, pages 347-377. Springer Berlin Heidelberg, 2003. URL: http://dx.doi.org/10.1007/978-3-642-55566-4_16.
10. D. Fon-Der-Flaass and A. V. Kostochka. Covering boxes by points. Disc. Math., 120(1-3):269-275, 1993. URL: http://dx.doi.org/10.1016/0012-365X(93)90587-J.
11. S. Govindarajan and G. Nivasch. A variant of the Hadwiger-Debrunner (p, q)-problem in the plane. Discrete & Computational Geometry, 54(3):637-646, 2015. URL: http://dx.doi.org/10.1007/s00454-015-9723-9.
12. A. Gyárfás and J. Lehel. Covering and coloring problems for relatives of intervals. Discrete Mathematics, 55(2):167-180, 1985. URL: http://dx.doi.org/10.1016/0012-365X(85)90045-7.
13. H. Hadwiger and H. Debrunner. Über eine variante zum Hellyschen satz. Archiv der Mathematik, 8(4):309-313, 1957. URL: http://dx.doi.org/10.1007/BF01898794.
14. H. Hadwiger and H. Debrunner. Combinatorial geometry in the plane. Translated by V. Klee. With a new chapter and other additional material supplied by the translator. Holt, Rinehart and Winston, New York, 1964.
15. Gy. Károlyi. On point covers of parallel rectangles. Period. Math. Hungar., 23(2):105-107, 1991. URL: https://doi-org.proxy1.athensams.net/10.1007/BF02280661.
16. C. Keller, S. Smorodinsky, and G. Tardos. On max-clique for intersection graphs of sets and the hadwiger-debrunner numbers. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, pages 2254-2263, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.148.
17. S. J. Kim, K. Nakprasit, M. J. Pelsmajer, and J. Skokan. Transversal numbers of translates of a convex body. Discrete Mathematics, 306(18):2166-2173, 2006. URL: http://dx.doi.org/10.1016/j.disc.2006.05.014.
18. D.J. Kleitman, A. Gyárfás, and G. Tóth. Convex sets in the plane with three of every four meeting. Combinatorica, 21(2):221-232, 2001. URL: http://dx.doi.org/10.1007/s004930100020.
19. D. Larman, J. Matoušek, J. Pach, and J. Töröcsik. A Ramsey-type result for planar convex sets. Bulletin of London Math. Soc., 26:132-136, 1994.
20. J. Matoušek. Bounded VC-dimension implies a fractional Helly theorem. Discrete & Computational Geometry, 31(2):251-255, 2004. URL: http://dx.doi.org/10.1007/s00454-003-2859-z.
21. R. Pinchasi. A note on smaller fractional Helly numbers. Discrete and Computational Geometry, 54(3):663-668, 2015.
22. N. Scheller. (p,q)-probleme für quaderfamilien. Master’s thesis, Universität Dortmund, 1996.
23. V. N. Vapnik and A. Ya. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264-280, 1971.
24. G. Wegner. Über eine kombinatorisch-geometrische Frage von Hadwiger und Debrunner. Israel J. Math., 3:187-198, 1965. URL: https://doi-org.proxy1.athensams.net/10.1007/BF03008396.