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dc.contributor.authorSah, Ashwin
dc.contributor.authorSawhney, Mehtaab
dc.date.accessioned2020-04-24T17:51:14Z
dc.date.available2020-04-24T17:51:14Z
dc.date.issued2019-07-12
dc.identifier.issn0305-0041
dc.identifier.issn1469-8064
dc.identifier.urihttps://hdl.handle.net/1721.1/124864
dc.description.abstractMay the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ 4−o(1) but not O(δ 4 ). Let M(δ) be the maximum number such that the following holds: for every ǫ > 0 and G = F n 2 with n sufficiently large, if A ⊆ G × G with A ≥ δ|G| 2 , then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” (x, y),(x + d, y),(x, y + d) ∈ A is at least (M(δ) − ǫ)|G| 2 . As a corollary via a recent result of Mandache, we conclude that M(δ) = δ 4−o(1) and M(δ) = ω(δ 4 ). On the other hand, for 0 < δ < 1/2 and sufficiently large N, there exists A ⊆ [N] 3 with |A| ≥ δN3 such that for every d 6= 0, the number of corners (x, y, z),(x + d, y, z),(x, y + d, z),(x, y, z + d) ∈ A is at most δ c log(1/δ)N 3 . A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3. ©2019en_US
dc.publisherCambridge University Press (CUP)en_US
dc.relation.isversionof10.1017/s0305004119000173en_US
dc.rightsCreative Commons Attribution-Noncommercial-Share Alikeen_US
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/4.0/en_US
dc.sourcearXiven_US
dc.subjectGeneral Mathematicsen_US
dc.titleTriforce and cornersen_US
dc.typeArticleen_US
dc.identifier.citationFox, Jacob, Ashwin Sah, Mehtaab Sawhney, David Stoner, and Yufei Zhao, "Triforce and corners." Mathematical proceedings of the Cambridge Philosophical Society 2019 (July 2019): p. 1-15 doi 10.1017/s0305004119000173 ©2019 Author(s)en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.relation.journalMathematical proceedings of the Cambridge Philosophical Societyen_US
dc.eprint.versionAuthor's final manuscripten_US
dc.type.urihttp://purl.org/eprint/type/JournalArticleen_US
eprint.statushttp://purl.org/eprint/status/PeerRevieweden_US
dspace.orderedauthorsJacob Fox; Ashwin Sah; Mehtaab Sawhney; David Stoner; Yufei Zhaoen_US
dspace.date.submission2019-11-24T15:03:54Z
mit.journal.volume2019en_US
mit.licenseOPEN_ACCESS_POLICY


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