Induced bisecting families for hypergraphs

TL;DR

This paper investigates the minimal size of a specific set of vectors in high-dimensional space that can orthogonally relate to all points in a hypercube, providing tight bounds for most cases.

Contribution

It establishes asymptotically tight bounds for the size of induced bisecting families in hypergraphs, advancing understanding of orthogonality relations in high-dimensional combinatorics.

Findings

Derived tight bounds for the minimal size of vector sets for most values of d.

Provided constructive upper bounds for the problem.

Identified limitations for even values of d in certain ranges.

Abstract

Two $n$-dimensional vectors $A$ and $B$, $A,B \in \mathbb{R}^n$, are said to be \emph{trivially orthogonal} if in every coordinate $i \in [n]$, at least one of $A(i)$ or $B(i)$ is zero. Given the $n$-dimensional Hamming cube $\{0,1\}^n$, we study the minimum cardinality of a set $\mathcal{V}$ of $n$-dimensional $\{-1,0,1\}$ vectors, each containing exactly $d$ non-zero entries, such that every `possible' point $A \in \{0,1\}^n$ in the Hamming cube has some $V \in \mathcal{V}$ which is orthogonal, but not trivially orthogonal, to $A$. We give asymptotically tight lower and (constructive) upper bounds for such a set $\mathcal{V}$ except for the even values of $d \in \Omega(n^{0.5+\epsilon})$, for any $\epsilon$, $0< \epsilon \leq 0.5$.