Induced bisecting families for hypergraphs

Two $n$-dimensional vectors $A$ and $B$, $A,B mathbb{R}^n$, are said to be emph{trivially orthogonal} if in every coordinate $i [n]$, 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 `possibleu0027 point $A {0,1}^n$ in the Hamming cube has some $V mathcal{V}$ 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 $d Omega(n^{0.5+epsilon})$, for any $epsilon$, $0u003c epsilon leq 0.5$.