In this paper, we address the following problem due to Frankl and Füredi (Discrete Math 50:323–328, 1984). What is the maximum number of hyperedges in an r-uniform hypergraph with n vertices, such that every set of \(r+1\) vertices contains 0 or exactly 2 hyperedges? They solved this problem for \(r=3\) . For \(r=4\) , a partial solution is given by Gunderson and Semeraro (J Comb Theory B 126:114–136, 2017). Assuming the existence of skew-symmetric conference matrices for every order divisible by 4, we give a solution for \(n\equiv 0,3\pmod {4}\) . We obtain these results by constructing 4-uniform hypergraphs via the diamonds of a tournament. Such a tournament is called the realization of the hypergraph. In this paper, we show that the problem of determining whether a 4-uniform hypergraph is realizable, can be reduced to the problem of deciding whether a 3-uniform hypergraph is realizable by the 3-cycles of a tournament.
