[This earlier MO question](https://mathoverflow.net/questions/25874/what-is-the-minimal-size-of-a-partial-order-that-is-universal-for-all-partial-or) asks to find the minimum size of a partial order that is universal for all partial orders of size $n$, i.e. any partial order of size $n$ embeds into it, preserving the order. In particular, the question asks if the minimum size $f(n)$ has a polynomial upper bound, to which the answer is no.

In this question, I am interested in some concrete values of $f(n)$ for small $n$.
So far, I know that:

- $f(0) = 0$

- $f(1) = 1$

- $f(2) = 3$

- $f(3) = 5$

- $f(4) = 8$

- $f(n) \ge 2n - 1$

- $f(n) \in \Omega(n^k)$ for all $k$

Can we compute some additional values in this sequence? In particular, can we compute $f(5)$?

### Notes

- I was able to verify $f(4) = 8$ [using a computer-assisted proof using a SAT solver](https://github.com/cdstanford/curiosities/blob/master/universal-poset/universal-poset.als). I also tried naive enumeration of posets and checking for universality, but this fails at around $f(4)$. Computing $f(5)$ may require smarter enumeration, in particular better symmetry breaking.

- The sequence does not appear to be in OEIS yet (it does not appear to be any of the sequences beginning with 1, 3, 5, 8). I submitted [this draft](https://oeis.org/draft/A337981), and it was suggested that the sequence should be posted to MathOverflow to find more terms.