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 \[|A|>\binom{ n+(k-1)q}{ (k-1)(q-1)}\] 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 $p>k$.
Categories: math.CO math.NT