  The tensor power of the clique on $t$ vertices (denoted by $K_t^n$) is the
graph on vertex set $\{1, ..., t\}^n$ such that two vertices $x, y \in \{1,
..., t\}^n$ are connected if and only if $x_i \neq y_i$ for all $i \in \{1,
..., n\}$. Let the density of a subset $S$ of $K_t^n$ to be $\mu(S) :=
\frac{|S|}{t^n}$, and let the vertex boundary of a set $S$ to be vertices which
are incident to some vertex of $S$, perhaps including points of $S$. We
investigate two similar problems on such graphs.
  First, we study the vertex isoperimetry problem. Given a density $\nu \in [0,
1]$ what is the smallest possible density of the vertex boundary of a subset of
$K_t^n$ of density $\nu$? Let $\Phi_t(\nu)$ be the infimum of these minimum
densities as $n \to \infty$. We find a recursive relation allows one to compute
$\Phi_t(\nu)$ in time polynomial to the number of desired bits of precision.
  Second, we study given an independent set $I \subseteq K_t^n$ of density
$\mu(I) = \frac{1}{t}(1-\epsilon)$, how close it is to a maximum-sized
independent set $J$ of density $\frac{1}{t}$. We show that this deviation
(measured by $\mu(I \setminus J)$) is at most $4\epsilon^{\frac{\log t}{\log t
- \log(t-1)}}$ as long as $\epsilon < 1 - \frac{3}{t} + \frac{2}{t^2}$. This
substantially improves on results of Alon, Dinur, Friedgut, and Sudakov (2004)
and Ghandehari and Hatami (2008) which had an $O(\epsilon)$ upper bound. We
also show the exponent $\frac{\log t}{\log t - \log(t-1)}$ is optimal assuming
$n$ tending to infinity and $\epsilon$ tending to $0$. The methods have
similarity to recent work by Ellis, Keller, and Lifshitz (2016) in the context
of Kneser graphs and other settings.
  The author hopes that these results have potential applications in hardness
of approximation, particularly in approximate graph coloring and independent
set problems.
