  It is a well known result due to Korshunov and Sapozhenko that the hypercube
in $n$ dimensions has $(1 + o(1)) \cdot 2 \sqrt e \cdot 2^{2^{n-1}}$
independent sets. Jenssen and Keevash investigated in depth Cartesian powers of
cycles of fixed even lengths far beyond counting independent sets. They wonder
to which extent their results extend to cycles of odd length, where not even
the easiest case, counting independent sets in Cartesian powers of the
triangle, is known. In this paper, we make progress on their question by
providing a lower bound, which we believe to be tight. We also obtain a less
precise lower bound for the number of independent sets in Cartesian powers of
arbitrary odd cycles and show how to approach this question both with the
cluster expansion method as well as more directly with isoperimetric
inequalities.
