{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T17:17:36Z","timestamp":1673025456983},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2010,3,19]],"date-time":"2010-03-19T00:00:00Z","timestamp":1268956800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2010,7]]},"abstract":"Let P<\/jats:italic> be a monotone increasing graph property, let G<\/jats:italic> = (V<\/jats:italic>, E<\/jats:italic>) be a graph, and let q<\/jats:italic> be a positive integer. In this paper, we study the (1: q<\/jats:italic>) Maker\u2013Breaker game, played on the edges of G<\/jats:italic>, in which Maker's goal is to build a graph that satisfies the property P<\/jats:italic>. It is clear that in order for Maker to have a chance of winning, G<\/jats:italic> itself must satisfy P<\/jats:italic>. We prove that if G<\/jats:italic> satisfies P<\/jats:italic> in some strong sense, that is, if one has to delete sufficiently many edges from G<\/jats:italic> in order to obtain a graph that does not satisfy P<\/jats:italic>, then Maker has a winning strategy for this game. We also consider a different notion of satisfying some property in a strong sense, which is motivated by a problem of Duffus, \u0141uczak and R\u00f6dl [6].<\/jats:p>","DOI":"10.1017\/s0963548310000064","type":"journal-article","created":{"date-parts":[[2010,3,19]],"date-time":"2010-03-19T08:58:36Z","timestamp":1268989116000},"page":"481-491","source":"Crossref","is-referenced-by-count":4,"title":["Playing to Retain the Advantage"],"prefix":"10.1017","volume":"19","author":[{"given":"NOGA","family":"ALON","sequence":"first","affiliation":[]},{"given":"DAN","family":"HEFETZ","sequence":"additional","affiliation":[]},{"given":"MICHAEL","family":"KRIVELEVICH","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2010,3,19]]},"reference":[{"key":"S0963548310000064_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01897304"},{"key":"S0963548310000064_ref7","doi-asserted-by":"publisher","DOI":"10.1137\/060654414"},{"key":"S0963548310000064_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0801-3_2"},{"key":"S0963548310000064_ref12","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-36.1.445"},{"key":"S0963548310000064_ref14","volume-title":"Introduction to Graph Theory","author":"West","year":"2001"},{"key":"S0963548310000064_ref6","first-page":"141","article-title":"Biased positional games on hypergraphs","volume":"34","author":"Duffus","year":"1998","journal-title":"Studia Sci. Math. Hungar."},{"key":"S0963548310000064_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s004930070002"},{"key":"S0963548310000064_ref2","first-page":"128","volume-title":"Proc. 37 ACM STOC","author":"Alon","year":"2005"},{"key":"S0963548310000064_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-5060(08)70335-2"},{"key":"S0963548310000064_ref8","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032718"},{"key":"S0963548310000064_ref9","unstructured":"[9] Johansson A. (1996) Asymptotic choice number for triangle free graphs. DIMACS Technical Report 91-95. 1196."},{"key":"S0963548310000064_ref10","first-page":"21","volume-title":"Proc. 4th Annual ACM\u2013SIAM Symposium on Discrete Algorithms","author":"Karger","year":"1993"},{"key":"S0963548310000064_ref11","doi-asserted-by":"publisher","DOI":"10.1137\/0112059"},{"key":"S0963548310000064_ref13","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-36.1.221"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548310000064","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,28]],"date-time":"2019-04-28T15:29:02Z","timestamp":1556465342000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548310000064\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,3,19]]},"references-count":14,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,7]]}},"alternative-id":["S0963548310000064"],"URL":"https:\/\/doi.org\/10.1017\/s0963548310000064","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,3,19]]}}}