  Denote by $F_5$ the $3$-uniform hypergraph on vertex set $\{1,2,3,4,5\}$ with
hyperedges $\{123,124,345\}$. Balogh, Butterfield, Hu, and Lenz proved that if
$p > K \log n / n$ for some large constant $K$, then every maximum $F_5$-free
subhypergraph of $G^3(n,p)$ is tripartite with high probability, and showed
that if $p_0 = 0.1\sqrt{\log n} / n$, then with high probability there exists a
maximum $F_5$-free subhypergraph of $G^3(n,p_0)$ that is not tripartite. In
this paper, we sharpen the upper bound to be best possible up to a constant
factor. We prove that if $p > C \sqrt{\log n} / n $ for some large constant
$C$, then every maximum $F_5$-free subhypergraph of $G^3(n, p)$ is tripartite
with high probability.
