An exact extremal result for tournaments and 4-uniform hypergraphs

In this paper, we address the following problem due to Frankl and F\"uredi (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 (2017) when $n=q+1$ for some prime power number $q\equiv3\pmod{4}$. Assuming the existence of skew-symmetric conference matrices for every order divisible by $4$, we give a solution for $n\equiv0\pmod{4}$ and for $n\equiv3\pmod{4}$.