Tur\'an Number of Generalized Triangles

The family ${\mathrm{\Sigma }}_r$ consists of all $r$-graphs with three edges $D_1,D_2,D_3$ such that $\mathrm{\mid }{D}_1\cap D_2\mathrm{\mid }=r-1$ and $D_1 \triangle D_2 \subseteq D_3$\mathrm{.}Ageneralizedtriangle,$$\mathcal{T}$_r \in \Sigma_r$isan$r$-graphon${1,2,\ldots,2r-1}$withthreeedges$D_1, D_2, D_3$,suchthat$D_1={1,2,\dots,r-1, r}, D_2= {1, 2, \dots, r-1, r+1 }$and$D_3 = {r, r+1, \dots, 2r-1}.$FranklandF\ddot{u}rediconjecturedthatforall$r\geq 4$,$ex(n,\Sigma_r) = ex(n,$\mathcal{T}$_r )$forallsufficientlylarge$n$ 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 $r=5,6$.