Suppose any two edges of $G$ have a non-empty intersection. Must $G$ contain $O(r^2)$ many vertices? Must there be two edges which meet in $\gg r$ many vertices?
Suppose any two edges of $G$ have a non-empty intersection. Must $G$ contain $O(r^2)$ many vertices? Must there be two edges which meet in $\gg r$ many vertices?
Erdős and Lovász [ErLo75] proved that there must be two edges which meet in $\gg \frac{r}{\log r}$ many vertices.
Alon has provided the following counterexample to the first question: as vertices take two sets $X$ and $Y$ of sizes $2r-2$ and $\frac{1}{2}\binom{2r-2}{r-1}$ respectively, where $Y$ corresponds to all partitions of $X$ into two equal parts. The edges are all subsets of $X$ of size $r$, and also all sets consisting of a subset of $X$ of size $r-1$ together with the unique element of $Y$ corresponding to the induced partition of $X$.
This hypergraph is intersecting, its chromatic number is $3$, and it has $\asymp 4^r/\sqrt{r}$ many vertices.