= Fano-plane Turán theorem
{c}
{title2=$\operatorname{ex}(n,F)=\binom n3-\binom{\lfloor n/2\rfloor}3-\binom{\lceil n/2\rceil}3$}
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.
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