Triforce and Corners

Abstract

May the $triforce$ be the 3-uniform hypergraph on six vertices with edges $\{123^{\prime },12^{\prime }3,1^{\prime }23\}$. We show that the minimum triforce density in a 3-uniform hypergraph of edge density $\delta$ is ${\delta }^{4-o(1)}$ but not $O({\delta }^4)$. Let $M(\delta )$ be the maximum number such that the following holds: for every $\epsilon >0$ and $G={\mathbb{F}}_2^n$ with $n$ sufficiently large, if $A\subseteq G\times G$ with $A\ge \delta |G{|}^2$, then there exists a nonzero "popular difference" $d\in G$ such that the number of "corners" $(x,y),(x+d,y),(x,y+d)\in A$ is at least $(M(\delta )-\epsilon )|G{|}^2$. As a corollary via a recent result of Mandache, we conclude that $M(\delta )={\delta }^{4-o(1)}$ and $M(\delta )=\omega ({\delta }^4)$. On the other hand, for $0<\delta <1/2$ and sufficiently large $N$, there exists $A\subseteq [N{]}^3$ with $|A|\ge \delta N^3$ such that for every $d\ne 0$, the number of corners $(x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+d)\in A$ is at most ${\delta }^{c\log (1/\delta )}{N}^3$. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.