Hereditary Discrepancies in Different Numbers of Colors II

TL;DR

This paper establishes a sharp bound relating the hereditary discrepancy of a hypergraph in two colors to its discrepancy in c colors, providing insights into color-based discrepancy measures.

Contribution

It introduces a tight bound connecting hereditary discrepancies in two and c colors, advancing understanding of hypergraph discrepancy relations.

Findings

Bound: disc(H,2)  K c disc(H,c)

The bound is proven to be sharp.

Provides a fundamental relation between two-color and c-color hereditary discrepancies.

Abstract

We bound the hereditary discrepancy of a hypergraph $\HH$ in two colors in terms of its hereditary discrepancy in $c$ colors. We show that $\herdisc(\HH,2) \le K c \herdisc(\HH,c)$, where $K$ is some absolute constant. This bound is sharp.