{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article10788","name":"From a (p,2)-Theorem to a Tight (p,q)-Theorem","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.\nWhile 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.\nIn 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).\nIn 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"],"author":[{"@type":"Person","name":"Keller, Chaya","givenName":"Chaya","familyName":"Keller"},{"@type":"Person","name":"Smorodinsky, Shakhar","givenName":"Shakhar","familyName":"Smorodinsky"}],"position":51,"pageStart":"51:1","pageEnd":"51:14","dateCreated":"2018-06-08","datePublished":"2018-06-08","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Keller, Chaya","givenName":"Chaya","familyName":"Keller"},{"@type":"Person","name":"Smorodinsky, Shakhar","givenName":"Shakhar","familyName":"Smorodinsky"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2018.51","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1016\/0001-8708(92)90052-M","http:\/\/dx.doi.org\/10.1007\/s00454-013-9559-0","http:\/\/dx.doi.org\/10.1007\/s00454-012-9417-5","http:\/\/dx.doi.org\/10.1007\/s00453-010-9410-4","http:\/\/dx.doi.org\/10.1007\/978-3-642-55566-4_16","http:\/\/dx.doi.org\/10.1016\/0012-365X(93)90587-J","http:\/\/dx.doi.org\/10.1007\/s00454-015-9723-9","http:\/\/dx.doi.org\/10.1016\/0012-365X(85)90045-7","http:\/\/dx.doi.org\/10.1007\/BF01898794","https:\/\/doi-org.proxy1.athensams.net\/10.1007\/BF02280661","http:\/\/dx.doi.org\/10.1137\/1.9781611974782.148","http:\/\/dx.doi.org\/10.1016\/j.disc.2006.05.014","http:\/\/dx.doi.org\/10.1007\/s004930100020","http:\/\/dx.doi.org\/10.1007\/s00454-003-2859-z","https:\/\/doi-org.proxy1.athensams.net\/10.1007\/BF03008396"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6302","volumeNumber":99,"name":"34th International Symposium on Computational Geometry (SoCG 2018)","dateCreated":"2018-06-08","datePublished":"2018-06-08","editor":[{"@type":"Person","name":"Speckmann, Bettina","givenName":"Bettina","familyName":"Speckmann"},{"@type":"Person","name":"T\u00f3th, Csaba D.","givenName":"Csaba D.","familyName":"T\u00f3th"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article10788","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6302"}}}