{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,11,15]],"date-time":"2024-11-15T14:40:15Z","timestamp":1731681615735,"version":"3.28.0"},"reference-count":14,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T00:00:00Z","timestamp":1720656000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T00:00:00Z","timestamp":1720656000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2024,12]]},"abstract":"Abstract<\/jats:title>For any integer $$h\\geqslant 2$$<\/jats:tex-math>\n \n h<\/mml:mi>\n \u2a7e<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, a set of integers $$B=\\{b_i\\}_{i\\in I}$$<\/jats:tex-math>\n \n B<\/mml:mi>\n =<\/mml:mo>\n \n \n {<\/mml:mo>\n \n b<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n \n i<\/mml:mi>\n \u2208<\/mml:mo>\n I<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a $$B_h$$<\/jats:tex-math>\n \n B<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-set if all h<\/jats:italic>-sums $$b_{i_1}+\\ldots +b_{i_h}$$<\/jats:tex-math>\n \n \n b<\/mml:mi>\n \n i<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:msub>\n +<\/mml:mo>\n \u2026<\/mml:mo>\n +<\/mml:mo>\n \n b<\/mml:mi>\n \n i<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with $$i_1<\\ldots <i_h$$<\/jats:tex-math>\n \n \n i<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <<\/mml:mo>\n \u2026<\/mml:mo>\n <<\/mml:mo>\n \n i<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are distinct. Answering a question of Alon and Erd\u0151s [2], for every $$h\\geqslant 2$$<\/jats:tex-math>\n \n h<\/mml:mi>\n \u2a7e<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> we construct a set of integers X<\/jats:italic> which is not a union of finitely many $$B_h$$<\/jats:tex-math>\n \n B<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-sets, yet any finite subset $$Y\\subseteq X$$<\/jats:tex-math>\n \n Y<\/mml:mi>\n \u2286<\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> contains an $$B_h$$<\/jats:tex-math>\n \n B<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-set Z<\/jats:italic> with $$|Z|\\geqslant \\varepsilon |Y|$$<\/jats:tex-math>\n \n |<\/mml:mo>\n Z<\/mml:mi>\n |<\/mml:mo>\n \u2a7e<\/mml:mo>\n \u03b5<\/mml:mi>\n |<\/mml:mo>\n Y<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where $$\\varepsilon :=\\varepsilon (h)$$<\/jats:tex-math>\n \n \u03b5<\/mml:mi>\n :<\/mml:mo>\n =<\/mml:mo>\n \u03b5<\/mml:mi>\n (<\/mml:mo>\n h<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We also discuss questions related to a problem of Pisier about the existence of a set A<\/jats:italic> with similar properties when replacing $$B_h$$<\/jats:tex-math>\n \n B<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-sets by the requirement that all finite sums $$\\sum _{j\\in J}b_j$$<\/jats:tex-math>\n \n \n \u2211<\/mml:mo>\n \n j<\/mml:mi>\n \u2208<\/mml:mo>\n J<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n b<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are distinct.<\/jats:p>","DOI":"10.1007\/s00493-024-00115-1","type":"journal-article","created":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T16:02:17Z","timestamp":1720713737000},"page":"1211-1232","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Pisier Type Theorems"],"prefix":"10.1007","volume":"44","author":[{"given":"Jaroslav","family":"Ne\u0161et\u0159il","sequence":"first","affiliation":[]},{"given":"Vojt\u011bch","family":"R\u00f6dl","sequence":"additional","affiliation":[]},{"given":"Marcelo","family":"Sales","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,7,11]]},"reference":[{"issue":"3\u20134","key":"115_CR1","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/BF02574042","volume":"13","author":"N Alon","year":"1995","unstructured":"Alon, N., Kalai, G.: Bounding the piercing number. Discret. Comput. Geom. 13(3\u20134), 245\u2013256 (1995)","journal-title":"Discret. Comput. Geom."},{"issue":"3","key":"115_CR2","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1016\/S0195-6698(85)80027-5","volume":"6","author":"N Alon","year":"1985","unstructured":"Alon, N., Erd\u0151s, P.: An application of graph theory to additive number theory. Eur. J. Comb. 6(3), 201\u2013203 (1985)","journal-title":"Eur. J. Comb."},{"issue":"1","key":"115_CR3","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/0001-8708(92)90052-M","volume":"96","author":"N Alon","year":"1992","unstructured":"Alon, N., Kleitman, D.J.: Piercing convex sets and the Hadwiger\u2013Debrunner $$(p, q)$$-problem. Adv. Math. 96(1), 103\u2013112 (1992)","journal-title":"Adv. Math."},{"issue":"1","key":"115_CR4","doi-asserted-by":"publisher","first-page":"137","DOI":"10.5802\/aif.1003","volume":"35","author":"J Bourgain","year":"1985","unstructured":"Bourgain, J.: Sidon sets and Riesz products. 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