Optimal labelling schemes for adjacency, comparability and reachability

We construct asymptotically optimal adjacency labelling schemes for every hereditary class containing $2^{\Omega(n^2)}$ $n$-vertex graphs as $n\to \infty$. This regime contains many classes of interest, for instance perfect graphs or comparability graphs, for which we obtain an efficient adjacency labelling scheme with labels of $n/4+o(n)$ bits per vertex. This implies the existence of a reachability labelling scheme for digraphs with labels of $n/4+o(n)$ bits per vertex. We also use this result to construct a poset on $2^{n/4+o(n)}$ elements, that contains all $n$-element posets. All these results are best possible, up to the lower order term, and solve several open problems in the area.