Turán Number of Generalized Triangles

Abstract: The family $\Sigma_r$ consists of all $r$-graphs with three edges $D_1,D_2,D_3$ such that $|D_1\cap D_2|=r-1$ and $D_1 \triangle D_2 \subseteq D_3$. A generalized triangle, $\mathcal{T}_r \in \Sigma_r$ is an $r$-graph on ${1,2,\ldots,2r-1}$ with three edges $D_1, D_2, D_3$, such that $D_1={1,2,\dots,r-1, r}, D_2= {1, 2, \dots, r-1, r+1 }$ and $D_3 = {r, r+1, \dots, 2r-1}.$ Frankl and F\"{u}redi conjectured that for all $r\geq 4$, $ex(n,\Sigma_r) = ex(n,\mathcal{T}_r )$ 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 $ex(n,\mathcal{T}_5)$ and $ex(n,\mathcal{T}_6)$ for sufficiently large $n$, proving the conjecture for $r=5,6$.