{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,4,29]],"date-time":"2024-04-29T13:17:50Z","timestamp":1714396670552},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2021,3,18]],"date-time":"2021-03-18T00:00:00Z","timestamp":1616025600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2021,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A family of vectors in [<jats:italic>k<\/jats:italic>]<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup> is said to be <jats:italic>intersecting<\/jats:italic> if any two of its elements agree on at least one coordinate. We prove, for fixed <jats:italic>k<\/jats:italic> \u2265 3, that the size of any intersecting subfamily of [<jats:italic>k<\/jats:italic>]<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup> invariant under a transitive group of symmetries is <jats:italic>o<\/jats:italic>(<jats:italic>k<\/jats:italic><jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>), which is in stark contrast to the case of the Boolean hypercube (where <jats:italic>k<\/jats:italic> = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.<\/jats:p>","DOI":"10.1017\/s0963548321000079","type":"journal-article","created":{"date-parts":[[2021,3,18]],"date-time":"2021-03-18T09:43:51Z","timestamp":1616060631000},"page":"899-904","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["On symmetric intersecting families of vectors"],"prefix":"10.1017","volume":"30","author":[{"given":"Sean","family":"Eberhard","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jeff","family":"Kahn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bhargav","family":"Narayanan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sophie","family":"Spirkl","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2021,3,18]]},"reference":[{"key":"S0963548321000079_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(89)80042-3"},{"key":"S0963548321000079_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/s004930050045"},{"key":"S0963548321000079_ref15","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-96-03732-X"},{"key":"S0963548321000079_ref20","unstructured":"[20] Margulis, G. A. , (1974) Probabilistic characteristics of graphs with large connectivity, Problemy Pereda\u010di Informacii 10 101\u2013108."},{"key":"S0963548321000079_ref4","first-page":"55","article-title":"The influence of variables in product spaces, Israel","volume":"77","author":"Bourgain","year":"1992","journal-title":"J. Math."},{"key":"S0963548321000079_ref22","doi-asserted-by":"crossref","unstructured":"[22] Pach, J. and Tardos, G. , (2015) Cross-intersecting families of vectors, Graphs Combin. 31 477\u2013495.","DOI":"10.1007\/s00373-015-1551-4"},{"key":"S0963548321000079_ref11","first-page":"252","article-title":"Erd\u0151s-Ko-Rado theorem with conditions on the maximal degree, J. Combin. Theory","volume":"46","author":"Frankl","year":"1987","journal-title":"Ser. 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Algebraic Discrete","volume":"1","author":"Frankl","year":"1980","journal-title":"Methods"},{"key":"S0963548321000079_ref17","doi-asserted-by":"crossref","unstructured":"[17] Hilton, A. J. W. and Milner, E. C. , (1967) Some intersection theorems for systems of finite sets, Quart. J. Math. Oxford, Series 2 18 369\u2013384.","DOI":"10.1093\/qmath\/18.1.369"},{"key":"S0963548321000079_ref8","doi-asserted-by":"publisher","DOI":"10.1090\/proc\/13452"},{"key":"S0963548321000079_ref3","doi-asserted-by":"crossref","unstructured":"[3] Berge, C. , (1974) Nombres de coloration de l\u2019hypergraphe h-parti complet, Hypergraph Seminar (Proc. First Working Sem., Ohio State Univ., Columbus, Ohio, 1972; dedicated to Arnold Ross), pp. 13\u201320. Lecture Notes in Math., Vol. 411.","DOI":"10.1007\/BFb0066175"},{"key":"S0963548321000079_ref2","unstructured":"[2] Babai, L. , Personal communication."},{"key":"S0963548321000079_ref16","doi-asserted-by":"publisher","DOI":"10.19086\/da.3103"},{"key":"S0963548321000079_ref6","first-page":"77","article-title":"Independent sets in graph powers are almost contained in juntas, Geom. Funct","volume":"18","author":"Dinur","year":"2008","journal-title":"Anal."},{"key":"S0963548321000079_ref24","first-page":"622","article-title":"Cross t-intersecting integer sequences from weighted Erd\u0151s-Ko-Rado, Combin. Probab","volume":"22","author":"Tokushige","year":"2013","journal-title":"Comput."},{"key":"S0963548321000079_ref1","doi-asserted-by":"publisher","DOI":"10.1006\/eujc.1995.0092"},{"key":"S0963548321000079_ref19","unstructured":"[19] Lubetzky, E. , (2018) Dynamics for the critical 2d potts\/fk model: many questions and a few answers, Charles River Lecture Series. https:\/\/cims.nyu.edu\/\u02dceyal\/talks\/charlesriver_potts.pdf."},{"key":"S0963548321000079_ref21","first-page":"386","article-title":"An analogue of the Erd\u0151s-Ko-Rado theorem for the Hamming schemes H(n, q), J. Combin. Theory","volume":"32","author":"Moon","year":"1982","journal-title":"Ser. 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