Smallest $n$ admitting an antichain on $[n]$ with $n-3$ distinct block sizes, each used $\ge r$ times (Erdős #776)

Fix $r\ge 2$. Consider families $A_1,\dots,A_m\subseteq\{1,\dots,n\}$ that form an *antichain* ($A_i\not\subseteq A_j$ for $i\neq j$) and satisfy the multiplicity condition: for every $t$, if some set has size $t$ then at least $r$ of the sets have size $t$. The number of *distinct sizes* used is at most $n-1$. Let $n_0(r)$ be the least $n$ for which there exists such a family using exactly $n-3$ distinct sizes. Determine $n_0(r)$, as a function of $r$, for small $r$.

Acceptance. FULLY RESOLVES (for a given small $r$): the exact threshold $n_0(r)$ — (a) an explicit antichain on $\{1,\dots,n_0(r)\}$ meeting the multiplicity condition and using exactly $n_0(r)-3$ distinct sizes (machine-verified antichain + size profile), and (b) a certificate (exhaustive/ILP infeasibility) that no admissible family exists for any $n<n_0(r)$. PARTIAL: verified upper bounds (an explicit admissible family for some $n$, giving $n_0(r)\le n$) or lower bounds (certified infeasibility for a specific $n$).