On An Extremal Hypergraph Problem Of Brown, Erdős And Sós

Let fr(n,v,e) denote the maximum number of edges in an r-uniform hypergraph on n vertices, which does not contain e edges spanned by v vertices. Extending previous results of Ruzsa and Szemerédi and of Erdős, Frankl and Rödl, we partially resolve a problem raised by Brown, Erdős and Sós in 1973, by showing that for any fixed 2≤kr, we have $$n^{{k - o{\left( 1 \right)}}}