An exact Turán result for the generalized triangle

AbstractLet Σk consist of all k-graphs with three edges D1, D2, D3 such that |D1 ∩ D2| = k − 1 and D1 Δ 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 $$ \{ 1, \ldots ,k\} , \{ 1,2, \ldots ,k - 1,k + 1\} , and \{ k,k + 1, \ldots ,2k - 1\} . $$ Frankl and Füredi [J. Combin. Theory Ser. (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.

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