Triforce和Corners
Triforce and Corners
摘要：
$\mathit{triforce}$可以是边为$\{123',12'3,1'23\}$的6点3一致超图.我们证明了边密度为$δ$的3-一致超图的最小三力密度是$δ^{4-o(1)}$而不是$O(δ^4)$。假设$M(δ)$是最大数，使得以下成立：对于每个$ε> 0$和$G = \mathbb{F}_2^n$，其中$n$足够大，如果$A \subseteq G \times G$与$A \ge δ|G|^2$，则存在非零的“流行差异”$d \in G$，使得“角”的数目$(x,y), (x+d,y), (x,y+d) \in A$至少为$(M(δ) - ε)|G|^2$。作为一个推论，通过最近的结果Mandache，我们得出结论，$M(δ) = δ^{4-o(1)}$和$M(δ) = ω(δ^4)$。另一方面，对于$0 < δ< 1/2$和足够大的$N$，存在具有$|A|\geδN^3$的$A \subseteq [N]^3$，使得对于每个$d \ne 0$，角的数目$(x,y,z), (x+d,y,z),(x,y+d,z),(x,y,z+d) \in A$至多为$δ^{c \log (1/δ)} N^3$。类似的界限在更高的维度中成立，或者对于至少有5个点或仿射维度至少为3的任何配置。
英文摘要：
May the $\mathit{triforce}$ 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 $ε> 0$ and $G = \mathbb{F}_2^n$ with $n$ sufficiently large, if $A \subseteq G \times G$ with $A \ge δ|G|^2$, 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 $(M(δ) - ε)|G|^2$. 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 $A \subseteq [N]^3$ with $|A|\geδN^3$ 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 $δ^{c \log (1/δ)} N^3$. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.
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