The Fano plane is the unique linear 3-uniform hypergraph on seven vertices and seven hyperedges. It is known that, for all $n \geq 8$, the balanced complete bipartite 3-uniform hypergraph on $n$ vertices, denoted by $B_n$, is the 3-uniform hypergraph on $n$ vertices with the largest number of hyperedges that does not contain a copy of the Fano plane. For sufficiently large $r$ and $n$, we show that $B_n$ admits the largest number of $r$-edge colorings with no rainbow copy of the Fano plane.

Mubayi and Pikhurko established several Turán-type results and stability results for $r$-uniform hypergraphs. In particular, they considered hypergraphs that avoid a copy of an expanded complete 2-graph and a copy of a Fan-hypergraph. Their Turán stability results tell us the following for some fixed families $\mathcal{F}$ of forbidden $r$-uniform subgraphs with Turán number ${ex}(n,\mathcal{F})$: for every $\delta>0$, there exist $\varepsilon>0$ and $n_0$ such that any $\mathcal{F}$-free $r$-uniform hypergraph with $n \geq n_0$ vertices and at least ${ex}(n,\mathcal{F})-\varepsilon n^r$ hyperedges gets the “structure” of an extremal hypergraph by removing at most $\delta n^r$ hyperedges. Here, we obtain sharper stability results. For some graph families $\mathcal{F}$, we find constants $a_\mathcal{F}$ and functions $b_\mathcal{F}=O(n^{r-1})$ and $p_\mathcal{F}=\Omega(n^r)$ such that any $n$-vertex $\mathcal{F}$-free $r$-uniform hypergraph with at least ${ex}(n,\mathcal{F})-p$ hyperedges, where $p