Random Turán Theorem for the Fano Plane

Let $F$ denote the Fano plane, the $3$-uniform hypergraph with $7$ vertices and $7$ edges. Frankl and F\"uredi, and independently Keevash and Sudakov, proved that the largest $F$-free subhypergraph of $K_n^{(3)}$ is bipartite. In this paper, we determine the sharp threshold for this property in the random setting. We show that for $\hat{p} = \Theta_F \cdot n^{-2/3} \left(\log n\right)^{1/6}$, where $\Theta_F$ is an explicit constant depending on $F$, we have: (i) if $(1+\epsilon) \hat{p} \le p = o(1)$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is bipartite; and (ii) if $\frac{1}{n^2} \ll p \le (1-\epsilon) \hat{p}$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is not bipartite. To the best of our knowledge, this work provides the first sharp threshold result obtained for a Tur\'an-type problem in random hypergraphs.