Title: System of unbiased representatives for a collection of bicolorings,
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 $\mathcal{A}$.