Independent sets in discrete tori of odd sidelength

TL;DR

This paper investigates the number of independent sets in Cartesian powers of odd cycles, providing new lower bounds and methods to approach the problem, extending previous work on even cycles and hypercubes.

Contribution

It offers the first lower bounds for independent sets in odd cycle powers and introduces approaches using cluster expansion and isoperimetric inequalities.

Findings

Established a tight lower bound for independent sets in odd cycle powers.

Provided a less precise lower bound for arbitrary odd cycles.

Demonstrated methods using cluster expansion and isoperimetric inequalities.

Abstract

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.