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From a (p,2)-Theorem to a Tight (p,q)-Theorem Keller, Chaya Smorodinsky, Shakhar (p,q)-Theorem convexity transversals (p,2)-theorem axis-parallel rectangles 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Chaya Keller and Shakhar Smorodinsky 2018 Is Part Of LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2018.51 urn:nbn:de:0030-drops-87640 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.51 eng https://creativecommons.org/licenses/by/3.0/legalcode