**The paper is published:**
Discrete Applied Mathematics

**MSC:**- 06A07 Combinatorics of partially ordered sets
- 11P82 Analytic theory of partitions

refinement. Let $b(Pi_n)$ be the largest size of a level and $d(Pi_n)$ be

the largest size of an antichain of $Pi_n$. We prove that

\[

\frac{d(Pi_n)}{b(Pi_n)} \le e + o(1) \mbox{ as } n \rightarrow \infty.

\]

The denominator is determined asymptotically. In addition, we show that the

incidence matrices in the lower half of $Pi_n$ have full rank, and we prove

a tight upper bound for the ratio from above if $Pi_n$ is replaced by any

graded poset $P$.