Reverse discrepancy and almost zero-sum stars

TL;DR

This paper investigates the maximum minimal imbalance at vertices in hypergraphs with edge functions summing to zero, providing exact solutions for complete and equipartite hypergraphs, thus contributing to hypergraph discrepancy theory.

Contribution

It introduces a reverse discrepancy problem for hypergraphs and derives exact results for specific hypergraph classes, expanding understanding of zero-sum configurations.

Findings

Exact maximum minimal imbalance values for complete hypergraphs

Exact maximum minimal imbalance values for equipartite hypergraphs

New insights into hypergraph discrepancy and zero-sum problems

Abstract

For $f$ chosen from the $\{-1,1\}$-valued functions on the edges of a hypergraph $\mathcal{H} = (V,E)$ with $\sum_{e \in E} f(e) = 0$, how large can one make $\min_{v \in V} |\sum_{e \ni v} f(e)|$? This question may be viewed as a reverse version of the hypergraph discrepancy problem or as a relaxation of the zero-sum Ramsey problem for stars. We prove exact results when $\mathcal{H}$ is a complete or equipartite hypergraph.