Judicious Partitions of 3-uniform Hypergraphs

A conjecture of Bollobás and Thomason asserts that, for r ≥ 1, everyr -uniform hypergraph with m edges can be partitioned into r classes such that every class meets at least rm / (2 r − 1) edges. Bollobás, Reed and Thomason proved that there is a partition in which every edge meets at least (1 − 1 / e)m / 3 ≈ 0.21 m edges. Our main aim is to improve this result for r = 3. We prove that every 3-uniform hypergraph with m edges can be partitioned into three classes, each of which meets at least (5 m − 1) / 9 edges. We also prove that for r > 3 we may demand 0.27 m edges.