Description
The sail (upside down the fan), $F$, has four triples on seven points, three pairwise
intersecting in the same point $p$ and the fourth intersecting all of them in points
different from $p$. More generally, the $k$-fan $F^k$ has $k$ $k$-sets pairwise intersecting in the same point $p$ and a crossing edge intersecting all of them in points different from $p$. Note that $F^2$ is the graph triangle and $F^3 = F$.
Linear hypergraph: any two edges intersect in at most one point. The linear Tur án
number of a linear hypergraph $L$, $ex_{lin}(n, L)$, is the maximum number of edges in a
linear hypergraph with $n$ points that does not contain $L$.
We proved (asked) with Zoli F ̈uredi (arXiv:1710.03042):
• 1. $ex_{lin}(n, F^k) ≤ n^2/k^2$, equality is possible only for transversal designs on
$n$ points with $k$ groups. The case $k = 2$ is known as Mantel’s theorem. In
fact, both ”book-proofs” (Aigner-Ziegler, Proofs from the BOOK) of Mantel’s
theorem extend nicely.
• 2. if $n = 3m+ 2$ then $ex_{lin}(n, F) = m^2 +m$, equality only in the following cases:
removing one point from a transversal design on $3m + 3$ points; extending each
factor of a factorization of a graph $G$ to triples, where $G$ is either the Wagner
graph ($C_8$ with long diagonals) or the graph $C_{5,2}$ (obtained from $C_5$ by doubling its points). The proof uses a special case of a result of Andr ásfai, Erdős and
Sós (1974).
• 3. if $n = 3m + 1$ then $ex_{lin}(n, F) =?$
Apart from $F$, there are two other linear triple systems having four triples on
at most seven points: $P$ (the Pasch configuration) and $C_{14 }. We can say something
about $ex_{lin}(n, \mathcal{A}) for all $\mathcal{A} \subseteq \{F, P, C_{14}\}$.