Triforce and corners

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