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 $s$，存在常数 $\gamma$ 和一个子图封闭类，使得对于任意自然数 $n$，该类中至多存在 $\gamma^n$ 个同构意义下的 $n$ 顶点图，但不存在标签大小不超过 $s \log n$ 的邻接标签方案。换言之，对于任意 $s$，存在一个极小（甚至微小）的单调类，该类不具有规模为 $n^s$ 的通用图。在此之前，并未排除每个小类别都具有近乎线性的通用图，或等价地具有标签大小为 $(1+o(1))\log n$ 的标签方案的可能性。这种标签方案作为近期被证伪的隐式图猜想（Implicit Graph Conjecture）的缩略版本，曾被多次提出 [Gavoille and Labourel, ESA '07; Dujmović et al., JACM '21; Bonamy et al., SIDMA '22; Bonnet et al., Comb. Theory '22]。此外，我们构造的小单调类具有无界双宽度，从而同时证伪了已被推翻的小猜想（Small conjecture）；但此次的证明是自包含的，且不依赖于复杂的群论构造。