1. System of unbiased representatives for a collection of bicolorings
- Author
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Balachandran, Niranjan, Mathew, Rogers, Mishra, Tapas Kumar, and Pal, Sudebkumar Prasant
- Subjects
Mathematics - Combinatorics ,Computer Science - Discrete Mathematics ,05Dxx, 05C65, 05C15, 68R05 ,G.2.1 ,G.2.2 - 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 $\mathcal{A}$., Comment: 14 pages, 1 figure
- Published
- 2017