Fix $n \geq 1$. For any finite collection $\mathcal{E}$ of closed convex subsets of $\mathbb{R}^n$, and for any $k \geq 1$, say that $\mathcal{E}$ has the $k$-intersection property if $$\bigcap_{i=1,\cdots,k}E_i \ne \emptyset \text{ for all } E_1,\cdots,E_k \in \mathcal{E}.$$
For fixed $n$, what is the largest $k$ for which there exists a collection that has the $k$-intersection property but not the $k+1$-intersection property?
For $n=1$ it is clear that the answer is $k=1$, as any finite collection $\mathcal{E}$ having the 2-intersection property will be such that $\bigcap_{E \in \mathcal{E}}E \ne \emptyset$, thereby implying the $k$-intersection property for any $k>2$. But I don't see how to approach the problem for larger $n$ (even for $n=2$).
This is Helly's theorem (with an obvious note that $n+1$ in its premise is the best possible).