Fano-plane Turán theorem ($\mathrm{ex}(n,F)=\left(\genfrac{}{}{0ex}{}{n}{3}\right)-\left(\genfrac{}{}{0ex}{}{\lfloor n/2\rfloor }{3}\right)-\left(\genfrac{}{}{0ex}{}{\lceil n/2\rceil }{3}\right)$)

For all sufficiently large $n$, the extremal number of the Fano plane as a three-uniform hypergraph is the displayed count of all triples crossing a balanced bipartition. The plane has no Property B, so this construction avoids it. A stability argument makes near-extremal Fano-free hypergraphs nearly bipartite; vertex deletion obtains high minimum degree. Optimizing the approximate bipartition and applying Fano completion through three matchings then rules out all internal triples. The large-order qualification matters: for $n=5,6$ the complete hypergraph has no Fano copy but exceeds the displayed count.

For every $\rho >0$, some $\delta >0$ makes every sufficiently large Fano-free three-uniform hypergraph of density at least $3/4-\delta$ admit a partition with at most $\rho n^3$ internal triples. The argument compares four hypergraph link graphs around a tetrahedron. Forbidden rainbow perfect matchings in their edge-coloured union force an approximate pattern with multiplicity two within the parts and four between them. A positive internal triple density would produce a complete tripartite six-vertex template, and its pairs together with a link vertex would complete a Fano plane.