Triforce and corners. (12th July 2019)

• Record Type: Journal Article
• Authors: FOX, JACOB; SAH, ASHWIN; SAWHNEY, MEHTAAB; STONER, DAVID; ZHAO, YUFEI
• Is Part Of: Mathematical proceedings of the Cambridge Philosophical Society. Volume 169:Part 1(2020)
• Page Start: 209
• Page End: 223
• Publication Date: 2019-07-12
• Subjects: 11B30, -- 05D05; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0305004119000173
• Languages: English
• ISSNs: 0305-0041

Abstract

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 ∊ > 0 and $G = {\mathbb{F}}_2^n$ 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 )–∊)| 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 | ≥ δN 3 such that for every d ≠ 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.