极值集合论，$\mathbb{F}_2^n$和Hurwitz平方恒等式上的三次型
Extremal set theory, cubic forms on $\mathbb{F}_2^n$ and Hurwitz square identities
摘要：
我们考虑一个家庭，$\mathcal{F}$，一个$n$集的子集，使任何两个元素的对称差的基数$F,F'\in\mathcal{F}$是不是4的倍数。我们证明了$\mathcal{F}$的最大尺寸是有界的$2n$，除非$n\equiv{}3\mod4$时，它是有界的$2n+2$。我们的方法使用$\mathbb{F}_2^n$上的三次型和Hurwitz-Radon平方恒等式理论。我们也应用这个理论来获得一些关于布尔三次型和所谓的加法四元组的信息。
英文摘要：
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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