On judicious partitions of hypergraphs with edges of size at most 3

Bollobás and Scott (2002) conjectured that a hypergraph with mi edges of size i for i=1,⋯,k has a bipartition in which each vertex class meets at least m1/2+3m2/4+⋯+(1−1/2k)mk+o(m) edges where m=∑i=1kmi. For the case k=2, this conjecture has been proved by Ma et al. (2010). In this paper, we consider this conjecture for the case k=3. In fact, we prove that a hypergraph with mi edges of size i for i=1,2,3 has a bipartition in which each vertex class meets at least m1/2+3m2/4+23m3/27+o(m) edges where m=m1+m2+m3.