The vertices of any graph with m edges may be partitioned into two parts so that each part meets at least 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 r/2r-1m 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 r/3r-4m edges, a substantial improvement on previous bounds.
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