Følner Sets from Variable-Free Terms

Definitions and mathematical statement

Let $\mathcal T$ be the set of formal variable-free terms generated by the symbol $1$ and a binary operation $\star$. Let $\sim$ be the congruence generated by the left-distributive identity \[ A\star(B\star C)\sim(A\star B)\star(A\star C). \] The supplied formulation stipulates a family of strictly increasing, nonidentity functions $f_T:{\mathbb{N}}\to{\mathbb{N}}$, indexed by $T\in\mathcal T$, satisfying \[ S\sim T\Longrightarrow f_S=f_T, \qquad \operatorname{crit}(f_{A\star B}) =f_A\left(\operatorname{crit}(f_B)\right), \qquad \{\operatorname{crit}(f_T):T\in\mathcal T\}={\mathbb{N}}. \] Under the usual interpretation, the critical point of such a function is \[ \operatorname{crit}(f)=\min\{k\in{\mathbb{N}}:f(k)\ne k\}. \] This interpretation is not explicitly defined in the supplied record and requires confirmation. The recursion below evaluates functions at $0$, so it requires $0\in{\mathbb{N}}$, or an explicitly supplied extension of their domains; the record does not state its convention for ${\mathbb{N}}$.

Define, on formal terms, \[ \phi(1)=0, \qquad \phi(A\star B)=\max\{\phi(A),f_A(\phi(B))\}. \] For positive integers $m$, write the right-associated powers as \[ R_1=1,\qquad R_{m+1}=1\star R_m. \] The original statement describes $A_{n,p}$ using iterated right divisors of $R_{p+1}$; the cleaned statement says only right divisors. Retaining the original qualification, its intended description is \[ A_{n,p} =\{T\in\operatorname{RDiv}_{\mathrm{it}}(R_{p+1}):\phi(T)\le n\}. \] Here $\operatorname{RDiv}_{\mathrm{it}}$ is a placeholder for the source's undefined iterated-right-divisor relation, not a completed mathematical definition. The record does not specify whether divisibility is taken among formal terms, modulo $\sim$, or in another algebraic object. If equivalence classes are intended, the use of $\phi$ on those classes also needs justification.

The proposed assertion, for the intended function family and intended divisor and Følner definitions, is \[ \forall\varepsilon\in{\mathbb{R}}_{>0}\ \exists p_0\in{\mathbb{N}}\ \forall p\in{\mathbb{N}}\ \left(p>p_0\Longrightarrow \exists n_0\in{\mathbb{N}}\ \forall n\in{\mathbb{N}}\ \left(n\ge n_0\Longrightarrow A_{n,p}\text{ is }\varepsilon\text{-F\o lner}\right)\right). \] Thus the finite exceptional set of indices $n$ may depend on both $\varepsilon$ and $p$. The supplied background calls these sets finite. However, the record supplies no ambient group, semigroup, or action, no transformations against which the Følner condition is tested, and no boundary inequality or normalization defining $\varepsilon$-Følner. Consequently this display preserves the proposed quantifiers but remains an incomplete formulation pending those definitions. The record also does not specify whether the assertion concerns one distinguished function family or every family satisfying the displayed conditions.

Short English statement

The question asks whether sufficiently large right-associated powers of the generator yield, through bounded-$\phi$ iterated right divisors, $\varepsilon$-Følner sets for every sufficiently large bound $n$. The threshold for $n$ may depend on $\varepsilon$ and the chosen power. The intended divisor relation, ambient action, precise Følner test, and function-family quantification are missing, so the supplied record does not determine a complete precise conjecture.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30001682$, title "Følner Sets from Variable-Free Terms," exact source code catalogue entry. The supplied record attributes the problem to the Oberwolfach Reports Open Problems collection and gives proposed year $2011$; no individual proposer is supplied.
• Set Theory (2011), Oberwolfach Reports, pp. 85--140, DOI 10.4171/owr/2011/02. Supplied report citation and literature-source label "Oberwolfach problem statement": https://doi.org/10.4171/owr/2011/02.
• Supplied literature-source label "current amenability survey": https://arxiv.org/abs/2305.07113. The record supplies no author, title, or passage identifying its relevance to the exact conjecture.
• Literature assessment date: 2026-08-21. The supplied dated curation labels the problem open and reports that no later proof of the exact conjectured test was found. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official report was checked and the binary operation, function system, critical-point identities, and Følner-set definition were restored. This is supplied verification metadata, not independent verification here; the supplied statement still omits an explicit Følner-set definition.