Relative discrepancy of hypergraphs

TL;DR

This paper investigates the relative discrepancy in hypergraphs, providing bounds and exact values for a key parameter, using advanced mathematical techniques to extend classical combinatorial results.

Contribution

It determines the exact values of the discrepancy parameter for hypergraphs with uniformity up to 13 and improves bounds for larger hypergraphs, answering open questions in the field.

Findings

Exact values of bs(k) for 2 ≤ k ≤ 13.

Upper bound bs(k)=O(k^{0.525}) for large k.

Extension of classical theorems to hypergraphs.

Abstract

Given $k$-uniform hypergraphs $G$ and $H$ on $n$ vertices with densities $p$ and $q$, their relative discrepancy is defined as $\hbox{disc}(G,H)=\max\big||E(G')\cap E(H')|-pq\binom{n}{k}\big|$, where the maximum ranges over all pairs $G',H'$ with $G'\cong G$, $H'\cong H$, and $V(G')=V(H')$. Let $\hbox{bs}(k)$ denote the smallest integer $m \ge 2$ such that any collection of $m$ $k$-uniform hypergraphs on $n$ vertices with moderate densities contains a pair $G,H$ for which $\hbox{disc}(G,H) = \Omega(n^{(k+1)/2})$. In this paper, we answer several questions raised by Bollob\'as and Scott, providing both upper and lower bounds for $\hbox{bs}(k)$. Consequently, we determine the exact value of $\hbox{bs}(k)$ for $2\le k\le 13$, and show $\hbox{bs}(k)=O(k^{0.525})$, substantially improving the previous bound $\hbox{bs}(k)\le k+1$ due to Bollob\'as-Scott. The case $k=2$ recovers a result of Bollob\'as-Scott, which generalises classical theorems of Erd\H{o}s-Spencer, and Erd\H{o}s-Goldberg-Pach-Spencer. The case $k=3$ also follows from the results of Bollob\'as-Scott and Kwan-Sudakov-Tran. Our proof combines linear algebra, Fourier analysis, and extremal hypergraph theory.