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Induced bisecting families for hypergraphs
- Publication Year :
- 2016
-
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 $\epsilon$, $0< \epsilon \leq 0.5$.<br />Comment: 9 pages, 1 figure
- Subjects :
- Mathematics - Combinatorics
05D05, 05C65
G.2.1
G.2.2
F.2.2
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1610.00140
- Document Type :
- Working Paper