co.combinatorics - What is the minimal size of a partial order that is universal for all partial orders of size n?

A partial order $mathbb{B}$ is universal for a class $cal{P}$ of partial orders if every order in $cal{P}$ embeds
order-preservingly into $mathbb{B}$.

For example, every partial order
$langlemathbb{P},ltrangle$ maps order-preservingly into
its power set by the map
$$pmapsto{qinmathbb{P}mid qleq p}$$ that sends each element $p$ to its
lower cone.

Thus, the power set order $langle
P({1,2,ldots,n}),{subseteq}rangle$ is universal for
the class of partial orders of size $n$. This provides an
order of size $2^n$ that is universal for orders of size
$n$.

Question. What is the minimal size of a partial
order that is universal for orders of size $n$?

In particular, is there a polynomial upper bound?

One can make at least slight improvements to the $2^n$
upper bound, by observing that the emptyset was not needed,
as it never arises as a lower cone, and we don't need all
the atoms, since if they are needed, then one can use the
co-atoms instead. I suspect that there is a lot of waste in
the power set order, but the best upper bound I know is
still exponential.

For a lower bound, my current knowledge is weak and far
from exponential. Any order that is universal for orders of
size $n$ will contain a chain and an antichain, making it
have size at least $2n-1$. (That bound is exact for $nleq
3$.) A student in my intro logic course extended this to
$nlog(n)$ by considering $k$ chains (and antichains) of size
$n/k$.

Can one find better lower bounds?

Interestingly, the same student observed that we cannot in
general expect to find unique smallest universal orders,
since he found several orders of size 5 that are
universal for orders of size 3 and which are minimal with
that property. So in general, we cannot expect a unique
optimal universal order. Does this phenomenon occur for
every $n$? (He also found minimal universal orders of size larger than the minimal size universal order.)