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

TL;DR

This paper investigates the maximum size of set families with symmetric differences not divisible by four, using cubic forms over finite fields and Hurwitz-Radon theory, revealing bounds dependent on the dimension modulo four.

Contribution

It introduces a novel approach combining cubic forms and Hurwitz-Radon theory to bound the size of specific set families, connecting combinatorics with algebraic structures.

Findings

Maximum size bounds for set families depending on n mod 4

Application of cubic forms to boolean functions and additive quadruples

Use of Hurwitz-Radon theory to derive combinatorial bounds

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.