Is there $n\in\mathbb{N}$ and a collection ${\cal C}$ of subsets of $\{1,\ldots,n\}$ with the following properties?


1. $|{\cal C}| = n$,
2. $|c| > 1$ for all $c\in {\cal C}$,
3. $c\neq d \in {\cal C} \implies |c\cap d|=1$, and
4. $\big|\{|c|: c\in {\cal C}\}\big| > 2$.



Intersecting subsets of $\{1,\ldots,n\}$
No, there isn't. This is essentially the dual version of the [De Bruijn-Erdos theorem](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(incidence_geometry)) if the elements of $\mathcal C$ are the points, and the elements from $\{1,\ldots,n\}$ are the lines. The original proof is [here](https://www.renyi.hu/~p_erdos/1948-01.pdf).


