Even smaller universal posets

We show that for every $\eta>0$ and sufficiently large $n$, there exists a poset of size $2^{(1+\eta)n/2}$ containing all the $n$-element posets as induced subposets.

This improves a recent result of Bastide, Groenland and Nenadov.

Our proof provides a labeling scheme preserving transitivity, inspired by the Boolean lattice.

Among other tools, we use the Szemerédi Regularity Lemma.