On 3-cube-free constructions in the integers modulo 2^n

Washock, Ethan

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On 3-cube-free constructions in the integers modulo 2^n

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Classical theorems in extremal combinatorics due to Sperner, Erd˝os, Kleitman, and Samotij state that families minimizing the amount of chains in a Boolean lattice are restricted to a “layered” construction. These theorems translate from the Boolean lattice to the integers modulo 2n when k-chains are replaced with projective cubes of dimension 2^{k−1} in the case of k being a power of two. This case was proven by Long and Wagner in 2018. Conjectured constructions of largest k-cube-free for any k are also conjectured in their paper, which also have a specific layered construction. However, these bounds on the size of a k-cube-free set aren’t proven. In this thesis, I will investigate the structure of 3-cube-free subsets of the integers modulo 2^n and derive strategies for bounding the “fullness” of layers in a 3-cube-free construction that could possibly be extended to deal with any k-cube-free set.

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