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

Bollobas and Scott (2002) conjectured that a hypergraph with m(i) edges of size i for i = 1, . . . , k has a bipartition in which each vertex class meets at least m(1)/2+3m(2)/4+ . . . +(1-1/2(k))m(k)+o(m) edges where m = Sigma(k)(i=1) m(i). 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 m(1)/2 + 3m(2)/4+ 23m(3)/27 + o(m) edges where m = m(1) + m(2) + m(3). (C) 2015 Elsevier Ltd. All rights reserved.