The family
Σr consists of all
r -graphs with three edges
033d3aa" title="Click to view the MathML source">D1,D2,D3 such that
|D1∩D2|=r−1 and
D1△D2⊆D3. A
generalized triangle ,
Tr∈Σr is an
r -graph on
{1,2,…,2r−1} with three edges
033d3aa" title="Click to view the MathML source">D1,D2,D3, such that
e3544872b83741e5f8c594470b6695f1" title="Click to view the MathML source">D1={1,2,…,r−1,r},D2={1,2,…,r−1,r+1} and
D3={r,r+1,…,2r−1}.
Frankl and Füredi conjectured that for all r≥4, e358ba" title="Click to view the MathML source">ex(n,Σr)=ex(n,Tr) for all sufficiently large n and they also proved it for 3ef129da" title="Click to view the MathML source">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.