The family
b14b9c582" title="Click to view the MathML source">Σr consists of all
r -graphs with three edges
b87d0c6b3f6e0b0033d3aa" title="Click to view the MathML source">D1,D2,D3 such that
88546b9e5c5456f1df1eae2adf784219" title="Click to view the MathML source">|D1∩D2|=r−1 and
88b07cc30535d3176f8d8c" title="Click to view the MathML source">D1△D2⊆D3. A
generalized triangle ,
Tr∈Σr is an
r -graph on
b81fba6d" title="Click to view the MathML source">{1,2,…,2r−1} with three edges
b87d0c6b3f6e0b0033d3aa" title="Click to view the MathML source">D1,D2,D3, such that
b83741e5f8c594470b6695f1" 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, e4883fc2a206ab82de358ba" title="Click to view the MathML source">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 8862af786bbf43" title="Click to view the MathML source">ex(n,T5) and e4d40610a130945" title="Click to view the MathML source">ex(n,T6) for sufficiently large n , proving the conjecture for b88e12330290f88" title="Click to view the MathML source">r=5,6.