On judicious partitions of uniform hypergraphs

Bollobás and Scott showed that the vertices of a graph of m edges can be partitioned into k   sets such that each set contains at most m/k2+o(m)m/k2+o(m) edges. They conjectured that the vertices of an r  -uniform hypergraph, where r≥3r≥3, of m edges may likewise be partitioned into k   sets such that each set contains at most m/kr+o(m)m/kr+o(m) edges. In this paper, we prove the weaker statement that a partition into k   sets can be found in which each set contains at most m(k−1)r+r1/2(k−1)r/2+o(m) edges. Some partial results on this conjecture are also given.