I am looking to find the upper bound for Turán numbers $t_3(11,5)$ and $t_3(16,7)$. Here $t_r(n,m)$ denotes the smallest integer $k$ such that each $r$-uniform hyper graph on $n$ vertices with $k+1$ edges must contain a complete graph on $m$ vertices. Observe that when $H'$ is a complete graph on $m$ vertices, we have $t_{r}(n,m)=t_{r}(n,H')$.
I am looking for any known results for bounds of the Turán numbers $t_3(11,5)$ and $t_3(16,7)$, but I cannot seem to find any.
There seems to be known results for the case of $t_3(n,4)$ (see https://www.sciencedirect.com/science/article/pii/S0097316598929612).
We tackle the complementary question: what is the minimum number $u_r(n,m)$ of $r$-subsets of an $n$-set $S$ so that every $m$-subset of $S$ contains at least one of the chosen $r$-subsets (i.e. all $m$-subsets are hit)? It is easy to see that $t_r(n,m)+u_r(n,m)=\binom nr$ and $u_r(n,m)=C(n,n-r,n-m)$, the fewest number of $n-r$-subsets of an $n$-set needed to cover each $n-m$-set at least once.
Using the covering equivalence and Klas Markström's page about them gives exact numbers for both parameter sets of interest: $$u_3(11,5)=C(11,8,6)=29\implies t_3(11,5)=\binom{11}3-29=136$$ $$u_3(16,7)=C(16,13,9)=39\implies t_3(16,7)=521$$ Furthermore, it gives that there is $1$ extremal graph for $t_3(11,5)$ up to isomorphism and $6$ for $t_3(16,7)$.