On Extremal $k$-Graphs Without Repeated Copies of 2-Intersecting Edges

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 $k=4$.