An Erdős--Trotter problem on antichains with multiplicity $r$ on each occurring level

Fix an integer $r\ge 2$. For each $n$ we consider families $\mathcal{F}\subseteq 2^{[n]}$ that form an antichain and have the property that, for every $t$, if there exists $A\in \mathcal{F}$ with $\mathrm{\mid }A\mathrm{\mid }=t$ then there exist at least $r$ members of $\mathcal{F}$ of size $t$. A problem of Erdős and Trotter asserts that, for each fixed $r$, there exists a threshold $n_0(r)$ such that whenever $n>n_0(r)$ one can achieve $n-3$ distinct set sizes in such a family, and asks for estimates on $n_0(r)$. We compute that $n_0(2)=3$ and $n_0(3)=8$. For all $r\ge 4$ we prove matching linear bounds up to lower-order terms, namely $$2r+2\le n_0(r)\le 2r+2{\log }_{2}r+O({\log }_2{\log }_{2}r)\mathrm{.}$$ In particular, $n_0(r)=2r+o(r)$.