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arXiv · 1704.07716

System of unbiased representatives for a collection of bicolorings

Niranjan Balachandran·Rogers Mathew·Tapas Kumar Mishra·Sudebkumar Prasant Pal
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Abstract

Let $\mathcal{B}$ denote a set of bicolorings of $[n]$, where each bicoloring is a mapping of the points in $[n]$ to $\{-1,+1\}$. For each $B \in \mathcal{B}$, let $Y_B=(B(1),\ldots,B(n))$. For each $A \subseteq [n]$, let $X_A \in \{0,1\}^n$ denote the incidence vector of $A$. A non-empty set $A$ is said to be an `unbiased representative' for a bicoloring $B \in \mathcal{B}$ if $\left\langle X_A,Y_B\right\rangle =0$. Given a set $\mathcal{B}$ of bicolorings, we study the minimum cardinality of a family $\mathcal{A}$ consisting of subsets of $[n]$ such that every bicoloring in $\mathcal{B}$ has an unbiased representative in $\mathcal{A}$.

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Niranjan Balachandran, Rogers Mathew, Tapas Kumar Mishra, Sudebkumar Prasant Pal. 2017-04-25. System of unbiased representatives for a collection of bicolorings. https://arxiv.org/abs/1704.07716

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