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.

翻译：我们证明对于任意自然数 $s$，存在一个常数 $\gamma$ 和一个子图封闭类，使得对于任意自然数 $n$，该类中顶点数为 $n$ 的图在同构意义下至多有 $\gamma^n$ 个，但不存在标签大小不超过 $s \log n$ 的邻接标签方案。换言之，对每个 $s$，存在一个微小（甚至极小）的单调类，其通用图的大小无法达到 $n^s$。在此结果之前，我们无法排除每个小类都具有几乎线性的通用图，或等价地具有标签大小为 $(1+o(1))\log n$ 的标号方案的可能性。此类标号方案（作为近期被证伪的隐式图猜想的缩减版本）曾被多次提出 [Gavoille 和 Labourel, ESA '07; Dujmović 等人, JACM '21; Bonamy 等人, SIDMA '22; Bonnet 等人, Comb. Theory '22]。此外，我们构造的小单调类具有无界孪生宽度，从而同时证伪了已被反驳的小猜想；但此次证明是自包含的，不依赖于复杂的群论构造。