Turán number of generalized triangles

Frankl and Füredi conjectured that for all r≥4, ex(n,Σr)=ex(n,Tr) 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,T5) and ex(n,T6) for sufficiently large n, proving the conjecture for r=5,6.