  The $k$-th $p$-power of a graph $G$ is the graph on the vertex set $V(G)^k$,
where two $k$-tuples are adjacent iff the number of their coordinates which are
adjacent in $G$ is not congruent to 0 modulo $p$. The clique number of powers
of $G$ is poly-logarithmic in the number of vertices, thus graphs with small
independence numbers in their $p$-powers do not contain large homogenous
subsets. We provide algebraic upper bounds for the asymptotic behavior of
independence numbers of such powers, settling a conjecture of Alon and Lubetzky
up to a factor of 2. For precise bounds on some graphs, we apply Delsarte's
linear programming bound and Hoffman's eigenvalue bound. Finally, we show that
for any nontrivial graph $G$, one can point out specific induced subgraphs of
large $p$-powers of $G$ with neither a large clique nor a large independent
set. We prove that the larger the Shannon capacity of $\bar{G}$ is, the larger
these subgraphs are, and if $G$ is the complete graph, then some $p$-power of
$G$ matches the bounds of the Frankl-Wilson Ramsey construction, and is in fact
a subgraph of a variant of that construction.
