The $O(m^{2/3})$ error term for judicious partitions of 3-uniform hypergraphs

Abstract

Let $k\ge2$ be a fixed integer. Bollob\'as and Scott proved that every $3$-uniform hypergraph with $m$ edges admits a partition of its vertex set into $k$ parts such that each part spans at most $m/k^3+O_k(m^{6/7})$ edges. Scott later suggested that the error term should be $O_k(m^{2/3})$. In this paper, we show that every $3$-uniform hypergraph with $m$ edges admits a partition into $k$ parts such that each part spans at most $m/k^3+O_k(m^{2/3})$ edges, thereby establishing the proposed error term.