I need help with the following:
What is the minimum $n$ such that the boolean lattice $2^{[n]}$ contains all posets of size $m$?
I noticed that it should contain a chain of length $m$, and the longest chain in $2^{[n]}$ has length $n+1$, so $n \ge m - 1$.
Also it should contain an antichain of size $m$, so $n \geq n'$, where $n'$ is the least number such that ${n' \choose \lfloor n'/2\rfloor} \geq m$. By manually checking we can see that for $m = 1,2,3,4$; $n$ is respectively $0,2,3,4$, and from there the first condition starts to imply the second, and my conjecture is that if $m \geq 5$ then $n = m-1$.
I appreciate any help trying to prove (or disprove) this. Thanks.
Hint: map each element of the poset to its down-set.