On the Erd{\H o}s--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 [RaiSh]. It is known 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 $m(2,r)=\left(\genfrac{}{}{0px}{}{r+1}{2}\right)$. 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.