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(δ⁴). Let M(δ) be the maximum number such that the following holds: for every ∊ > 0 and G = F₂ⁿ 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(δ⁴). 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 ≠ 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)}N³. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3. [ABSTRACT FROM AUTHOR]