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- W2894895242 abstract "In the Boolean lattice, Sperner's, Erdős's, Kleitman's and Samotij's theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of $mathbb{Z}_2^n$ we work in $mathbb{Z}_{2^n}$, several analogous statements hold if one replaces the word $k$-chain by projective cube of dimension $2^{k-1}$. We say that $B_d$ is a projective cube of dimension $d$ if there are numbers $a_1, a_2, ldots, a_d$ such that $$B_d = left{sum_{iin I} a_i biggrvert emptyset neq Isubseteq [d]right}.$$ As an analog of Sperner's and Erdős's theorems, we show that whenever $d=2^{ell}$ is a power of two, the largest $d$-cube free set in $mathbb{Z}_{2^n}$ is the union of the largest $ell$ layers. As an analog of Kleitman's theorem, Samotij and Sudakov asked whether among subsets of $mathbb{Z}_{2^n}$ of given size $M$, the sets that minimize the number of Schur triples (2-cubes) are those that are obtained by filling up the largest layers consecutively. We prove the first non-trivial case where $M=2^{n-1}+1$, and conjecture that the analog of Samotij's theorem also holds. Several open questions and conjectures are also given." @default.
- W2894895242 created "2018-10-12" @default.
- W2894895242 creator A5036535417 @default.
- W2894895242 creator A5074413461 @default.
- W2894895242 date "2018-10-02" @default.
- W2894895242 modified "2023-09-27" @default.
- W2894895242 title "The largest projective cube-free subsets of $mathbb{Z}_{2^n}$" @default.
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