An exact Turán result for the generalized triangle

Let Σk consist of all k-graphs with three edges D1, D2, D3 such that |D1∩D2| = k − 1 and D1 △ D2 ⊆ D3. The exact value of the Turán function ex(n,Σk) was computed for k = 3 by Bollobás [Discrete Math., 8 (1974) 21–24] and for k = 4 by Sidorenko [Math Notes, 41 (1987) 247–259]. Let the k-graph Tk ∈ Σk have edges {1, . . . , k}, {1, 2, . . . , k − 1, k + 1}, and {k, k + 1, . . . , 2k − 1}. Frankl and Füredi [J. Combin. Theory (A), 52 (1989) 129–147] conjectured that there is n0 = n0(k) such that ex(n, Tk) = ex(n,Σk) for all n ≥ n0 and had previously proved this for k = 3 in [Combinatorica, 3 (1983) 341–349]. Here we settle the case k = 4 of the conjecture. Abbreviated title: Generalized Triangle MSC: Primary 05D05, Secondary 05C35 ∗Partially supported by the National Science Foundation, Grant DMS-0457512.