On An Extremal Hypergraph Problem Of Brown, Erdős And Sós

Abstract

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 Szemeredi and of Erdős, Frankl and Rodl, we partially resolve a problem raised by Brown, Erdős and Sos in 1973, by showing that for any fixed 2≤k<r, we have $$n^{{k - o{\left( 1 \right)}}} < f_{r} {\left( {n,3{\left( {r - k} \right)} + k + 1,3} \right)} = o{\left( {n^{k} } \right)}.$$