The Exact Linear Turán Number of the Sail

A hypergraph is linear if any two of its edges intersect in at most one vertex. The sail (or 3-fan) F³ is the 3-uniform linear hypergraph consisting of 3 edges f₁, f₂, f₃ pairwise intersecting in the same vertex v and an additional edge g intersecting each f_i in a vertex different from v. The linear Turán number ex_lin(n, F³) is the maximum number of edges in a 3-uniform linear hypergraph on n vertices that does not contain a copy of F³. Füredi and Gyárfás proved that if n = 3k, then ex_lin(n, F³) = k² and the only extremal hypergraphs in this case are transversal designs. They also showed that if n = 3k + 2, then ex_lin(n, F³) = k² + k, and the only extremal hypergraphs are truncated designs (which are obtained from a transversal design on 3k + 3 vertices with 3 groups by removing one vertex and all the hyperedges containing it) along with three other small hypergraphs. However, the case when n = 3k + 1 was left open. In this paper, we solve this remaining case by proving that ex_lin(n, F³) = k² + 1 if n = 3k + 1, answering a question of Füredi and Gyárfás. We also characterize all the extremal hypergraphs. The difficulty of this case is due to the fact that these extremal examples are rather non-standard. In particular, they are not derived from transversal designs like in the other cases.