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arXiv · 1304.0949

Extremal set theory, cubic forms on $\mathbb{F}_2^n$ and Hurwitz square identities

Sophie Morier-Genoud·Valentin Ovsienko
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Abstract

We consider a family, $\mathcal{F}$, of subsets of an $n$-set such that the cardinality of the symmetric difference of any two elements $F,F'\in\mathcal{F}$ is not a multiple of 4. We prove that the maximal size of $\mathcal{F}$ is bounded by $2n$, unless $n\equiv{}3\mod4$ when it is bounded by $2n+2$. Our method uses cubic forms on $\mathbb{F}_2^n$ and the Hurwitz-Radon theory of square identities. We also apply this theory to obtain some information about boolean cubic forms and so-called additive quadruples.

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Sophie Morier-Genoud, Valentin Ovsienko. 2014-03-27. Extremal set theory, cubic forms on $\mathbb{F}_2^n$ and Hurwitz square identities. https://arxiv.org/abs/1304.0949

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