 
Summary: On an Extremal Hypergraph Problem of Brown, Erdos and SŽos
Noga Alon
Asaf Shapira
Abstract
Let fr(n, v, e) denote the maximum number of edges in an runiform hypergraph on n vertices,
which does not contain e edges spanned by v vertices. Extending previous results of Ruzsa and
SzemerŽedi and of Erdos, Frankl and Ršodl, we partially resolve a problem raised by Brown, Erdos
and SŽos in 1973, by showing that for any fixed 2 k < r, we have
nko(1)
< fr(n, 3(r  k) + k + 1, 3) = o(nk
).
1 Introduction
All the hypergraphs considered here are finite and have no parallel edges. An runiform hypergraph
(=rgraph for short) H = (V, E), is a hypergraph in which each edge contains precisely r distinct
vertices of V . Denote by fr(n, v, e) the largest number of edges in an rgraph on n vertices that
contains no e edges spanned by v vertices. Estimating the asymptotic growth of this function for fixed
integers r, e and v is one of the most well studied problems in extremal graph theory. In particular,
when e = v
r we get the well known TurŽan problem of determining the maximum possible number
of edges in an rgraph that contains no complete rgraph on v vertices. See the surveys [11], [8] and
