Partitioning 3-uniform hypergraphs

Bollobás and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges can be partitioned into r sets so that each set meets at least rm/(2r−1) edges. For r=3, Bollobás, Reed and Thomason proved the lower bound (1−1/e)m/3≈0.21m, which was improved to (5/9)m by Bollobás and Scott and to 0.6m by Haslegrave. In this paper, we show that any 3-uniform hypergraph with m edges can be partitioned into 3 sets, each of which meets at least 0.65m−o(m) edges.