Description
We call a $3$-graph with vertex set $\{a,b,c,d,e\}$ and edge set $\{abc,abd,cde\}$ a {\it generalized triangle}, denoted by $F_5$. In 1983, Frankl and F\"{u}redi proved that the maximum number of edges in an $F_5$-free $3$-graph is $\lfloor \frac{n}{3}\rfloor\lfloor \frac{n+1}{3}\rfloor\lfloor \frac{n+2}{3}\rfloor$ for $n\geq 3000$. In this paper, we determine the maximum number of edges in an $F_5$-free $3$-graph that is not tripartite for $n\geq 5000$. It implies that every $F_5$-free $3$-graph $\mathcal{H}$ with more than $\lfloor \frac{n}{3}\rfloor\lfloor \frac{n+1}{3}\rfloor\lfloor \frac{n+2}{3}\rfloor-\lfloor \frac{n-3}{3}\rfloor\lfloor \frac{n+1}{3}\rfloor$ edges must be 3-partite.
This work is joint with Jialu Shan, Jian Wang and Weihua Yang.