# Intersecting subsets of $\{1,\ldots,n\}$

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. $$\big|\{|c|: c\in {\cal C}\}\big| > 2$$.

No, there isn't. This is essentially the dual version of the De Bruijn-Erdos theorem if the elements of $$\mathcal C$$ are the points, and the elements from $$\{1,\ldots,n\}$$ are the lines. The original proof is here.