SOLVED
Does there exist an absolute constant $c>0$ such that, for all $r\geq 2$, in any $r$-uniform hypergraph with chromatic number $3$ there is a vertex contained in at least $(1+c)^r$ many edges?
In general, determine the largest integer $f(r)$ such that every $r$-uniform hypergraph with chromatic number $3$ has a vertex contained in at least $f(r)$ many edges. It is easy to see that $f(2)=2$ and $f(3)=3$. Erdős did not know the value of $f(4)$.
This was solved by Erdős and Lovász [ErLo75], who proved in particular that there is a vertex contained in at least
\[\frac{2^{r-1}}{4r}\]
many edges.
