Triforce and corners
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...
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| Format: | Article |
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Cambridge University Press (CUP)
2020
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| Online Access: | https://hdl.handle.net/1721.1/124864 |
| _version_ | 1811097812931182592 |
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| author | Sah, Ashwin Sawhney, Mehtaab |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Sah, Ashwin Sawhney, Mehtaab |
| author_sort | Sah, Ashwin |
| collection | MIT |
| description | 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 |
| first_indexed | 2024-09-23T17:05:21Z |
| format | Article |
| id | mit-1721.1/124864 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T17:05:21Z |
| publishDate | 2020 |
| publisher | Cambridge University Press (CUP) |
| record_format | dspace |
| spelling | mit-1721.1/1248642022-10-03T10:21:20Z Triforce and corners Sah, Ashwin Sawhney, Mehtaab Massachusetts Institute of Technology. Department of Mathematics General Mathematics 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 2020-04-24T17:51:14Z 2020-04-24T17:51:14Z 2019-07-12 Article http://purl.org/eprint/type/JournalArticle 0305-0041 1469-8064 https://hdl.handle.net/1721.1/124864 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) 10.1017/s0305004119000173 Mathematical proceedings of the Cambridge Philosophical Society Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Cambridge University Press (CUP) arXiv |
| spellingShingle | General Mathematics Sah, Ashwin Sawhney, Mehtaab Triforce and corners |
| title | Triforce and corners |
| title_full | Triforce and corners |
| title_fullStr | Triforce and corners |
| title_full_unstemmed | Triforce and corners |
| title_short | Triforce and corners |
| title_sort | triforce and corners |
| topic | General Mathematics |
| url | https://hdl.handle.net/1721.1/124864 |
| work_keys_str_mv | AT sahashwin triforceandcorners AT sawhneymehtaab triforceandcorners |