# On small $n$-uniform hypergraphs with positive discrepancy

## TL;DR

This paper investigates the minimal size of n-uniform hypergraphs that lack a zero-discrepancy two-coloring, refining existing bounds by establishing an upper bound proportional to the logarithm of the smallest non-divisor of n.

## Contribution

The authors improve the upper bound on the minimum number of edges in such hypergraphs, relating it to the logarithm of the smallest non-divisor of n, advancing understanding of discrepancy in hypergraphs.

## Key findings

- Established an upper bound: f(n) ≤ c log snd(n)
- Refined previous bounds on hypergraph discrepancy
- Connected hypergraph properties to number-theoretic functions

## Abstract

A two-coloring of the vertices $V$ of the hypergraph $H=(V, E)$ by red and blue has discrepancy $d$ if $d$ is the largest difference between the number of red and blue points in any edge. Let $f(n)$ be the fewest number of edges in an $n$-uniform hypergraph without a coloring with discrepancy $0$. Erd\H{o}s and S\'os asked: is $f(n)$ unbounded? N. Alon, D. J. Kleitman, C. Pomerance, M. Saks and P. Seymour proved upper and lower bounds in terms of the smallest non-divisor ($\mbox{snd}$) of $n$. We refine the upper bound as follows: $$f (n) \leq c \log \mbox{snd}\ {n}.$$