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

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 $F,F^{\prime }\in \mathcal{F}$ is not a multiple of 4. We prove that the maximal size of $\mathcal{F}$ is bounded by $2n$, unless $n\equiv 3\mathrm{mod}4$ 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.