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We construct asymptotically optimal adjacency labelling schemes for everyhereditary class containing $2^{\Omega(n^2)}$ $n$-vertex graphs as $n\to\infty$. This regime contains many classes of interest, for instance perfectgraphs or comparability graphs, for which we obtain an adjacency labellingscheme with labels of $n/4+o(n)$ bits per vertex. This implies the existence ofa reachability labelling scheme for digraphs with labels of $n/4+o(n)$ bits pervertex and comparability labelling scheme for posets with labels of $n/4+o(n)$bits per element. All these results are best possible, up to the lower orderterm.

Optimal labelling schemes for adjacency, comparability, and reachability