论文标题
$ k $二维无角的网格子集的数量
The number of $k$-dimensional corner-free subsets of grids
论文作者
论文摘要
$ k $ dimensional网格$ \ {1,2，\ cdots，n \}^k $的子集$ a $如果不包含表单$ \ {a \ \} \ cup \ cup \ a + de_ \ de_ \ leq iq k k yq k k y q i q k $ d \ leq k k y q y q y q y q i q k $ d \ leq iq k k y q y quye，则称为$ k $ dimensional corter-free。 \ {1,2，\ cdots，n \}^k $和$ d> 0 $，其中$ e_1，e_2，\ cdots，e_k $是$ \ mathbb {r}^k $的标准基础。我们定义了$ \ {1,2，\ cdots，n \}^k $的最大尺寸为$二维的无角子集，$ c_k（n）$。在本文中，我们表明$ k $二维无角的子集的数量是$ k $二维网格$ \ {1,2，\ cdots，n \}^k $最多最多是$ 2^{o（c_k（n））$，对于无限的$ n $ n $ n $ n $ n $ n $ n $ n $。我们的证明的主要工具是$ k $维的尺寸$θ（c_k（n））$和HyperGraph容器方法的超饱和结果。
A subset $A$ of the $k$-dimensional grid $\{1,2, \cdots, N\}^k$ is called $k$-dimensional corner-free if it does not contain a set of points of the form $\{ a \} \cup \{ a + de_i : 1 \leq i \leq k \}$ for some $a \in \{1,2, \cdots, N\}^k$ and $d > 0$, where $e_1,e_2, \cdots, e_k$ is the standard basis of $\mathbb{R}^k$. We define the maximum size of a $k$-dimensional corner-free subset of $\{1,2, \cdots, N\}^k$ by $c_k(N)$. In this paper, we show that the number of $k$-dimensional corner-free subsets of the $k$-dimensional grid $\{1,2, \cdots, N\}^k$ is at most $2^{O(c_k(N))}$ for infinitely many values of $N$. Our main tool for the proof is a supersaturation result for $k$-dimensional corners in sets of size $Θ(c_k(N))$ and the hypergraph container method.
