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We show that for any natural number $s$, there is a constant $\gamma$ and asubgraph-closed class having, for any natural $n$, at most $\gamma^n$ graphs on$n$ vertices up to isomorphism, but no adjacency labeling scheme with labels ofsize at most $s \log n$. In other words, for every $s$, there is a small (eventiny) monotone class without universal graphs of size $n^s$. Prior to thisresult, it was not excluded that every small class has an almost linearuniversal graph, or equivalently a labeling scheme with labels of size$(1+o(1))\log n$. The existence of such a labeling scheme, a scaled-downversion of the recently disproved Implicit Graph Conjecture, was repeatedlyraised [Gavoille and Labourel, ESA '07; Dujmovi\'{c} et al., JACM '21; Bonamyet al., SIDMA '22; Bonnet et al., Comb. Theory '22]. Furthermore, our smallmonotone classes have unbounded twin-width, thus simultaneously disprove thealready-refuted Small conjecture; but this time with a self-contained proof,not relying on elaborate group-theoretic constructions.

Small But Unwieldy: A Lower Bound on Adjacency Labels for Small Classes