An exact Turán result for tripartite 3-graphs

Abstract: Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let $K_4^-={123,124,134}$, $F_6={123,124,345,156}$ and $\mathcal{F}={K_4^-,F_6}$: for $n\neq 5$ the unique $\mathcal{F}$-free 3-graph of order $n$ and maximum size is the balanced complete tripartite 3-graph $S_3(n)$ (for $n=5$ it is $C_5^{(3)}={123,234,345,145,125}$). This extends an old result of Bollob\'as that $S_3(n) $ is the unique 3-graph of maximum size with no copy of $K_4^-={123,124,134}$ or $F_5={123,124,345}$.