Bounds for pairs in partitions of graphs

In this paper we study the following problem of Bollobas and Scott: What is the smallest f(k,m) such that for any integer k>=2 and any graph G with m edges, there is a partition V(G)=@?"i"="1^kV"i such that for 1@?i j@?k, e(V"i@?V"j)@?f(k,m)? We show that f(k,m) =23. (While the graph K"1","n shows that f(k,m)>=m/(k-1), which is 1.5m/k when k=3.) We also show that f(4,m)@?m/3+o(m) and f(5,m)@?4m/15+o(m), providing evidence to a conjecture of Bollobas and Scott. For dense graphs, we improve the bound to 4m/k^2+o(m), which, for large graphs, answers in the affirmative a related question of Bollobas and Scott.