# System of unbiased representatives for a collection of bicolorings

## TL;DR

This paper investigates the minimal size of a family of subsets of [n] such that each bicoloring in a given set has an unbiased representative, contributing to combinatorial design theory.

## Contribution

It introduces the concept of unbiased representatives for bicolorings and analyzes the minimal family size needed for coverage.

## Key findings

- Established bounds on the minimum size of such families.
- Characterized conditions for the existence of unbiased representatives.
- Provided algorithms for constructing small unbiased representative families.

## 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 $\mathcal{A}$.