Title: The Partition Rank of a Tensor and $k$-Right Corners in $\mathbb{F}_{q}^{n}$,
Abstract: Following the breakthrough of Croot, Lev, and Pach, Tao 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, and prove
that any set $A\subset\mathbb{F}_{q}^{n}$ of size
\[|A|>(k+1)\cdot\binom{n+(k-1)q}{(k-1)(q-1)}\] 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 $p>k$.