On several partitioning problems of Bollobás and Scott

Judicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. Let G be a hypergraph with m"i edges of size i for i=1,2. We show that for any integer k>=1, V(G) admits a partition into k sets each containing at most m"1/k+m"2/k^2+o(m"2) edges, establishing a conjecture of Bollobas and Scott. We also prove that V(G) admits a partition into k>=3 sets, each meeting at least m"1/k+m"2/(k-1)+o(m"2) edges, which, for large graphs, implies a conjecture of Bollobas and Scott (the conjecture is for all graphs). For k=2, we prove that V(G) admits a partition into two sets each meeting at least m"1/2+3m"2/4+o(m"2) edges, which solves a special case of a more general problem of Bollobas and Scott.

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