Turán number of generalized triangles

Abstract

The family Σr consists of all r-graphs with three edges 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 D1,D2,D3, such that 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, 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.