Triforce and corners
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1903.04863.pdf
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Author(s)
Sah, Ashwin
Sawhney, Mehtaab
Date Issued
July 12, 2019
Journal
Mathematical proceedings of the Cambridge Philosophical Society
Publisher
Cambridge University Press (CUP)
Citation
Fox, 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)
Version
Author's final manuscript
Abstract
May 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. ©2019
Subjects
General Mathematics
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1017/s0305004119000173
