On the Erdős–Hajnal problem in the case of $3$-graphs

Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. For the broad history of the problem see [10]. It is known [4] that for a fixed $n$ the sequence
$$ \frac{m(n,r)}{r^n} $$
has a limit. The only trivial case is $n=2$ in which $m(2,r) = \binom{r+1}{2}$. In this note we focus on the case $n=3$. First, we compare the existing methods in this case and then improve the lower bound.