Graded discrepancy of graphs and hypergraphs

Abstract: Let $G$ be a graph with $n$ vertices and $p\binom{n}{2}$ edges. We prove that there is an ordering $v_1, \ldots, v_n$ of the vertices in $G$ such that $|e({v_1, \ldots, v_\ell})-p\binom{\ell}{2}|\le c(p)(n-1)$ for all $\ell\in{1,\ldots,n}$, where $\min{p,\sqrt{p}-p}-o(p)\le c(p)\le\max{p,1-p}$. This solves an open problem suggested by Bollob\'as and Scott. We also extend this result to the hypergraph setting.