Following the breakthrough of Croot, Lev, and Pach [4], Tao [10] introduced a symmetrized version of their argument, which is now known as the slice rank method. In this paper, we introduce a more general version of the slice rank of a tensor, which we call the Partition Rank. This allows us to extend the slice rank method to problems that require the variables to be distinct. Using the partition rank, we generalize a recent result of Ge and Shangguan [6], and prove that any set $A\subset {\mathbb{F}}_q^n$ of size$|A|>\left(\begin{array}{c}n+(k-1)q\\ (k-1)(q-1)\end{array}\right)$ contains a k-right-corner, that is distinct vectors $x_1,\dots ,x_k,x_{k+1}$ where $x_1-x_{k+1},\dots ,x_k-x_{k+1}$ are mutually orthogonal, for $q=p^r$, a prime power with $p>k$.

随着Croot，Lev和Pach [4]的突破，Tao [10]引入了他们的参数的对称形式，现在称为切片秩方法。在本文中，我们介绍了张量的切片等级的更一般的版本，我们称之为分区等级。这使我们可以将切片等级方法扩展到要求变量不同的问题。利用划分等级，我们推广了葛和上官[6]的最新结果，并证明了任何集$\mathrm{一种}\subset {\mathbb{F}}_q^{ñ}$ 大小$|\mathrm{一种}|>（\begin{array}{c}ñ+（ķ-\mathrm{1个}）q\\ （ķ-\mathrm{1个}）（q-\mathrm{1个}）\end{array}）$包含k-right-corner，这是不同的向量$X_{\mathrm{1个}}，\dots ，X_{ķ}，X_{ķ+\mathrm{1个}}$ 哪里 $X_{\mathrm{1个}}-X_{ķ+\mathrm{1个}}，\dots ，X_{ķ}-X_{ķ+\mathrm{1个}}$ 互相正交 $q=p^{\mathrm{[R}}$，具有 $p>ķ$。