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Structures with not too fast unlabelled growth

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper classifies every countable structure whose unlabelled orbit growth stays below $2^n/p(n)$ for all polynomials $p$, showing that such groups are built from finite symmetries layered over the rational order.

desk verdict A substantial and likely correct classification of the class S, with real consequences, but the written proof has a repairable gap in Theorem 4.4 and leans heavily on an unpublished source. read the letter →

arxiv 2507.16985 v2 pith:PMQN3PQD submitted 2025-07-22 math.LO math.GR

Bertalan Bodor This is my paper
classification math.LOmath.GR MSC 03C4503C3520B27
keywords unlabelledgrowtholigomorphicpermutationgroupsomega-categoricalstructuresThomas'conjecturefinitecoversofhighlyset-transitivehereditarilycellularinterpretabilityin(Q<)
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Let $\mathscr{S}$ be the class of countable structures whose number $u_n$ of orbits on $n$-element subsets never reaches $2^n/p(n)$ for any polynomial $p$; the paper gives a complete classification of the automorphism groups of these structures. The classification says that every such group is assembled, by finite direct products, wreath products with $\mathrm{Sym}(\omega)$, finite-index supergroups, and isomorphisms, from copies of $\mathrm{Aut}(\mathbb{Q};<)$ acting on finite highly set-transitive fibres — so the rational order is the only infinite primitive ingredient. From this description the paper derives that $u_n$ grows like $\gamma_d^n$ for exactly one of the numbers $\gamma_d$ defined as the largest real root of $x^d-x^{d-1}-\cdots-1$, with the sequence increasing to $2$. It follows that there are only countably many such structures up to bidefinability, that each is first-order interpretable in $(\mathbb{Q};<)$ and is interdefinable with a finitely bounded homogeneous structure, and that each has finitely many first-order reducts — Thomas' conjecture for $\mathscr{S}$.

What carries the argument

The central object is the cover construction $L(\tilde{G}, G^*, D)$: starting from a linked finite cover $\tilde{G}$ of a hereditarily cellular group $G^*$, one attaches to each orbit of $G^*$ a datum $D(a) = (F_a, B_a, \phi_a)$ consisting of a fibre group $F_a$, a normal pointwise binding group $B_a$, and a surjection onto the fibre of $\tilde{G}$, and the resulting group consists of all permutations whose coordinate actions lie in the prescribed fibre data and are witnessed by an element of $\tilde{G}$. This construction captures exactly the covers with finite fibre factors. The second load-bearing mechanism is the ladder of classes $\mathscr{S}_d$ with thresholds $\gamma_d$, the largest real roots of $x^d - x^{d-1} - \cdots - 1$; the recurrence $u_n = \sum_{i=1}^{|F|} u_i(H) u_{n-i}(G_0)$ for wreath products $H \wr \mathrm{Aut}(\mathbb{Q};<)$ forces the growth constant to be one of the $\gamma_d$. Finite highly set-transitive groups $H$ are the only finite ingredients needed, and the rational order supplies the unique infinite primitive behaviour.

What would settle it

Take any closed oligomorphic permutation group $G$ whose unlabelled growth satisfies $u_n(G) \sim c^n$ for a real $c$ not lying in $\{\gamma_d : d \in \mathbb{N}\}$, while still $u_n(G) < 2^n/p(n)$ for every polynomial $p$; if such a group exists, the gap theorem (Theorem 1.5) is false. A natural place to look is among wreath products of finite highly set-transitive groups over structures interpretable in $(\mathbb{Q};<)$, since the classification predicts that every growth constant appearing in $\mathscr{S}$ must be one of the $\gamma_d$.

Watch

Extended reading notes

Core claim

The central claim is Theorem 7.10: for $d \in \mathbb{N} \cup \{\infty\}$, a permutation group $G$ lies in the class $\mathscr{S}_d$ of groups with unlabelled growth below $(\gamma_d+\varepsilon)^n$ if and only if $G$ is isomorphic to a group $L(\tilde{G}, G^*, D)$ constructed from finite covers of highly set-transitive groups over a hereditarily cellular base, and equivalently if and only if $G$ is built from $\mathrm{id}(\{\emptyset\})$ and groups $H \wr \mathrm{Aut}(\mathbb{Q};<)$, where $H$ is a finite highly set-transitive group of degree at most $d$, by isomorphisms, closed supergroups, finite direct products, and wreath products with $\mathrm{Sym}(\omega)$. Corollary 7.11 then gives $\mathscr{S} = \bigcup_d \mathscr{S}_d$, so every structure in $\mathscr{S}$ has exponential growth with base strictly below $2$. From the classification the paper derives the gap theorem (Theorem 1.5), interpretability in $(\mathbb{Q};<)$, finite homogenizability and finite boundedness, and Thomas' conjecture for $\mathscr{S}$.

Load-bearing premise

The load-bearing premise is the imported structural lemma saying that every structure in $\mathscr{S}$ has a cover whose base is hereditarily cellular and whose non-trivial fibres are finite covers of highly set-transitive, but not highly transitive, groups; the paper relies on this lemma to enter the classification, and if it fails for some $G \in \mathscr{S}$, Theorem 7.10 and its consequences collapse.

Editorial extensions

If this is right

  • Every structure in $\mathscr{S}$ has unlabelled growth whose $n$-th root tends to one of the numbers $\gamma_d$; no intermediate growth constants occur in this class.
  • The class $\mathscr{S}$ contains only countably many structures up to bidefinability.
  • Every structure in $\mathscr{S}$ is first-order interpretable in $(\mathbb{Q};<)$ and is interdefinable with a finitely bounded homogeneous structure.
  • Thomas' conjecture holds for $\mathscr{S}$: every structure in $\mathscr{S}$ has finitely many first-order reducts up to interdefinability.
  • A structure has at most polynomial unlabelled growth exactly when an expansion by finitely many constants is bidefinable with a finite disjoint union of copies of $(\mathbb{Q};<)$ together with a cellular structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the classification suggests that the exponential growth bases appearing among oligomorphic structures form a discrete ladder $1, \gamma_2, \gamma_3, \ldots \to 2$, so a structure whose growth constant is not one of these values cannot lie in $\mathscr{S}$; the paper does not claim such a statement outside $\mathscr{S}$.
  • Editorial inference: for constraint satisfaction, Theorem 1.6 reduces structures with polynomial unlabelled growth to combinations of $(\mathbb{Q};<)$ and cellular structures, but the paper notes complexity questions require primitive-positive definability, so the classification alone does not settle the infinite-domain CSP dichotomy for this class.
  • Editorial inference: a natural testable extension is to ask whether the same $L(\tilde{G},G^*,D)$ construction still classifies groups with unlabelled growth bounded by $c^n$ for a fixed $c<2$, rather than by $2^n/p(n)$ for every polynomial $p$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper classifies the class S of countable structures whose unlabelled orbit growth is not at least 2^n/p(n) for any polynomial p. The main theorem, Theorem 7.10, characterizes automorphism groups in S as builds from finite highly set-transitive groups, finite covers of reducts of (Q;<), and hereditarily cellular groups, using finite covers, wreath products with Sym(omega), finite direct products, and finite-index or closed supergroups. From this classification the paper derives a gap theorem for growth rates (Theorem 1.5), finite homogenizability and finite boundedness, interpretability in (Q;<), countability up to bidefinability, and Thomas' conjecture for S (Theorem 9.11).

Significance. If the proof is completed, this is a substantial contribution: it supplies a full structural description in the c<2 range of unlabelled growth, confirming two conjectures from Braunfeld and generalizing the classifications in [FT20] and [Bod24]. The paper is careful about what is imported from earlier work, and the main classification is concrete and falsifiable rather than an existence result. The theorem gives explicit sufficient conditions for membership in S_d, and the growth gap theorem is a sharp quantitative consequence. These strengths make the paper worth publishing, provided the load-bearing gap identified below is repaired.

major comments (2)
  1. [Section 4, Theorem 4.4] The stabilization argument applies Lemma 4.2 (= Corollary 5.3) to the sequence (G_{A_i,{C}})|C. Lemma 4.2 is a statement about the class F of closed finite covers of highly set-transitive groups. Immediately after Lemma 4.1 the paper explicitly notes that closedness of these restrictions does not follow from [Sim18b], and no argument is supplied before Theorem 4.4 showing that each (G_{A_i,{C}})|C is closed. Without closedness, an infinite descending chain of pairwise isomorphic non-closed groups with the same orbit partitions need not stabilize, since their closures can be equal while the groups themselves descend properly. Because Theorem 4.4 is what converts the cover from Lemma 4.1 into one with finite fiber factors, and Theorem 7.10, Theorem 1.5, and Theorem 9.11 all rely on that conversion, this is a load-bearing gap. The proof needs either a proof that the restrictions are closed, or a version of Lemma 4.2 for finite covers not assumed closed.
  2. [Section 6, Lemma 6.1] The proof asserts that the lifted triple (K,∇,∆) is an omega-partition of G and says this is clear from the definition, but condition (5) of Definition 2.18 — that G((C))/∆ = Sym(C/∆) for every ∇-class C — is not verified. This condition is needed for the induction step and hence for Lemma 6.3 and for the description of finite covers in Theorem 7.10(2). The gap is likely repairable by observing that the quotient G/∆ maps onto G*/∆*, but as written the verification is omitted and should be supplied explicitly.
minor comments (5)
  1. [Definition 2.16] The recursive definition appears internally inconsistent: H_{-1}=∅ makes H_0 empty by the recursion, while Remark 2.17 says H_0 is exactly the class of finite-degree groups. The base case of the recursion should be corrected.
  2. [Theorem 4.4 proof] The proof cites 'Lemma 5.9' before that lemma is stated; it should cite Lemma 4.3, which is the version stated earlier.
  3. [Lemma 7.7 and Theorem 7.10] Both proofs refer to 'Lemma 8', which does not exist in the manuscript; presumably Lemma 5.21 is intended.
  4. [Lemma 5.20] The polynomial is written f(x)=x^n - Σ_{i=1}^{k-1} c_i x^i, which does not match the displayed recursion an+k = Σ_{i=0}^{k-1} c_i an+i; the exponents in the polynomial should match the order k of the recurrence.
  5. [Title and heading] The title contains a typo ('f ast'), and the heading 'F acts 2.44' should read 'Facts 2.44'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the classification is derived from independent prior results, with only a terse closure step in Theorem 4.4 that is at most an expositional gap.

full rationale

The central derivation is not circular. The classification of S is obtained by importing the decomposition lemma from Simon [Sim18b] and the independently published classification of hereditarily cellular groups from the author's prior work [Bod24]; these are external inputs with stated assumptions that do not include the target classification, and they are not fitted to the present paper's conclusions. The growth computations in Sections 5 and 7 (e.g., Lemmas 5.19-5.21 and Theorem 7.10) are parameter-free and are checked against external benchmarks such as the five reducts of (Q;<) and the existing polynomial-growth classification. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the author's own work is invoked to forbid alternatives. The only circularity-adjacent point is the step in Theorem 4.4 where Lemma 4.2 (a statement about the class F of closed finite covers) is applied to the restriction groups (G_{A_i,{C}})|C, which are explicitly said only to be finite covers of highly set-transitive groups. This is a missing explicit closedness justification rather than a reduction of the theorem's conclusion to its hypothesis: closedness of restrictions of closed groups to invariant sets is a standard independent fact, and the paper's omission is an expositional gap. Self-citations to [Bod24] supply the base class H and are not load-bearing in a way that makes the argument self-referential.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted; the classification is built from structural data such as subgroups H, L, M of Sym(F) and finite covers rather than numerical constants. The main external postulates are prior classification theorems, chiefly [Sim18b] Lemma 4.1, [Bod24]'s hereditary cellular classification, Ivanov's finite-cover classification, Cameron's list of highly set-transitive groups, and the finite highly set-transitive group classification.

assumptions (7)
  • standard math Engeler-Ryll-Nardzewski-Svenonius theorem equating omega-categoricity with oligomorphicity of the automorphism group
    Used in Section 2.3 to move freely between structures and closed oligomorphic permutation groups.
  • domain assumption Cameron's classification of closed highly set-transitive groups as the automorphism groups of (Q;<) and its four reducts
    Theorem 2.41 is the base case for all finite-cover classifications in Section 5; the paper cites [Cam76].
  • domain assumption Ivanov's classification of finite covers of reducts of (Q;<), giving the normal form K_L^H(id(F) wreath Aut(Q;R)) and generators for Betw, Cyc and Sep
    Theorem 5.2 imports this directly from [Iva99], Section 2; it drives the classification of finite covers in Section 5.
  • domain assumption Simon's decomposition lemma (Lemma 4.1): every G in S has a cover over a hereditarily cellular group whose non-trivial fibers are finite covers of highly set-transitive but not highly transitive groups
    This is the main external input, taken from [Sim18b] without reproducing its proof; it opens the route to Theorem 4.4 and Theorem 7.10.
  • domain assumption Lachlan-Bod24 classification of hereditarily cellular structures as monadically stable omega-categorical structures, with closure under finite direct products, wreath products with the pure set, and finite-index supergroups
    The class H is the base of the classification; the paper cites [Lac92] and its own [Bod24], Theorems 2.19-2.22 and 2.27.
  • domain assumption Finite highly set-transitive groups are exactly Sym(n), Alt(n), AGL(1,5), PGL(2,8), and PGammaL(2,8)
    Theorem 5.23 from [LW65] restricts the possible fiber groups in the F intersect S classification.
  • domain assumption Falque and Thiéry's classification of P-oligomorphic groups as quasi-polynomial growth
    Used in Section 8 and in Theorem 8.2 and Theorem 1.6 to characterize polynomial unlabelled growth; cited as [FT20].

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Pith. "Pith review of Structures with not too fast unlabelled growth." pith.science (2026). https://pith.science/paper/PMQN3PQD

@misc{pith2026250716985,
  author       = {Pith},
  title        = {Pith review of: Structures with not too fast unlabelled growth},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PMQN3PQD}},
  note         = {Machine review of arXiv:2507.16985}
}
abstract

Let $\mathscr{S}$ be the class of all structures whose growth rate on orbits of subsets of size $n$ is not faster than $\frac{2^n}{p(n)}$ for any polynomial $p$. In this article we give a complete classification of all structures in $\mathscr{S}$ in terms of their automorphism groups. As a consequence of our classification we show that $\mathscr{S}$ has only countably many structures up to bidefinability, all these structures are first-order interpretable in $(\mathbb{Q};<)$ and they are interdefinable with a finitely bounded homogeneous structure. Furthermore, we also show that all structures in $\mathscr{S}$ have finitely many first-order reduct up to interdefinability, thereby confirming Thomas' conjecture for the class $\mathscr{S}$.

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