  The problem of determining extremal hypergraphs containing at most r
isomorphic copies of some element of a given hypergraph family was first
studied by Boros et al. in 2001. There are not many hypergraph families for
which exact results are known concerning the size of the corresponding extremal
hypergraphs, except for those equivalent to the classical Turan numbers. In
this paper, we determine the size of extremal k-uniform hypergraphs containing
at most one pair of 2-intersecting edges for k in {3,4}. We give a complete
solution when k=3 and an almost complete solution (with eleven exceptions) when
k=4.
