Judicious partitions of uniform hypergraphs

Abstract

The vertices of any graph with m edges may be partitioned into two parts so that each part meets at least $\tfrac{{2m}} {3}$ edges. Bollobás and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges may likewise be partitioned into r classes such that each part meets at least $\tfrac{r} {{2r - 1}}$ edges. In this paper we prove the weaker statement that, for each r ≥ 4, a partition into r classes may be found in which each class meets at least $\tfrac{r} {{3r - 4}}$ edges, a substantial improvement on previous bounds.