  We show that for any natural number $s$, there is a constant $\gamma$ and a
subgraph-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 of
size at most $s \log n$. In other words, for every $s$, there is a small (even
tiny) monotone class without universal graphs of size $n^s$. Prior to this
result, it was not excluded that every small class has an almost linear
universal graph, or equivalently a labeling scheme with labels of size
$(1+o(1))\log n$. The existence of such a labeling scheme, a scaled-down
version of the recently disproved Implicit Graph Conjecture, was repeatedly
raised [Gavoille and Labourel, ESA '07; Dujmovi\'{c} et al., JACM '21; Bonamy
et al., SIDMA '22; Bonnet et al., Comb. Theory '22]. Furthermore, our small
monotone classes have unbounded twin-width, thus simultaneously disprove the
already-refuted Small conjecture; but this time with a self-contained proof,
not relying on elaborate group-theoretic constructions.
