Turan number of generalized triangles

The family Er consists of all r-graphs with three edges D-1, D-2, D-3 such that vertical bar D-1 boolean AND D-2 vertical bar = r - 1 and D-1 Delta D-2 subset of D-3. A generalized triangle tau(r) is an element of Sigma(r) is an r-graph on {1, 2,, 2r - 1} with three edges D-1, D-2, D-3, such that D-1 = {1, 2, . . ., r - 1, r}, D-2 = {1, 2,..., r - 1, r+1} and D-3 = {r, r+1,..., 2r-1}. Frankl and Fiiredi conjectured that for all r >= 4, ex(n, Sigma(r)) = ex(n, tau(r)) for all sufficiently large n and they also proved it for r = 3. Later, Pikhurko showed that the conjecture holds for r = 4. In this paper we determine ex(n, tau(5)) and ex(n, tau(6)) for sufficiently large n, proving the conjecture for r = 5,6. (C) 2016 Published by Elsevier Inc.