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

Independent sets in discrete tori of odd sidelength