REVIEW 6 minor 6 references
Counterexamples to Problem 17.102 of the Kourovka Notebook: Negation and Discussion of Separability Conditions in Infinite Groups
T0 review · 0 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The paper proves that in some infinite groups, two disjoint subsets smaller than the group cannot be separated by any infinite symmetric set, refuting a conjecture from the Kourovka Notebook.
desk verdict A correct, clean refutation of Kourovka 17.102, anchored by a self-contained graph-theoretic counterexample; only typos and an odd AI disclosure keep it from being immediately publishable. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The witness set $W_G(A,B)$ is the central object: $u\in W$ exactly when $\{u,u^{-1}\}A\{u,u^{-1}\}$ avoids $B$. Its defining property is used twice: Lemma 3.1 shows every candidate separating set $X$ must be a subset of $W$, so finite $W$ is an obstruction; Lemma 3.2 then constructs an increasing chain $X_0\subset X_1\subset\cdots$ inside $W$, excluding at most $4|X_k||A||B|$ forbidden elements at each step, to produce a countable separating set when $|W|$ is large. The counterexamples are designed around this dichotomy: in the first family, the conjugation structure of a central element of order two restricts $W$ to two elements; in the second, a graph with trivial automorphism group makes $W=\{id\}$.
What would settle it
The graph-based counterexample is testable: verify that $W_G(A,B)=\{id\}$ by checking whether any non-identity permutation of $\mathbb{N}$ fails to move some edge transposition of the rigid graph to a non-edge transposition; if even one non-identity element lies in $W$, the claimed obstruction must be revisited.
Extended reading notes
Core claim
The paper's central discovery is that Conjecture 1.1 is false, and that separability is governed by a witness set $W_G(A,B)$, not by the cardinalities of $A$ and $B$ alone. In Theorem 1.1 it proves that if $|W_G(A,B)| > \max(|A|,|B|,\aleph_0)$ then a countable separating set exists, while if $W_G(A,B)$ is finite then no separating set exists, because any admissible $X$ must be contained in $W_G(A,B)$. Three families of examples are given: a product $H \times C_2$ where $H$ is torsion-free with $|H|=2^{\aleph_0}$ and $|H^2|=\aleph_0$, which gives $W=\{e,a\}$; a symmetric group $\mathrm{Sym}(\mathbb{N})$ with transpositions from a rigid graph's edges and non-edges, which gives $W=\{id\}$; and two further constructions where $W$ is countably infinite but $X$ is forced to be $\{e\}$. These examples establish the falsity of the conjecture and show the strict inequality in the sufficient condition is sharp.
Load-bearing premise
The load-bearing premise is that the explicit constructions really do have the stated witness sets—especially that the first family inherits the previously established existence of a torsion-free group with $|H|=2^{\aleph_0}$ and $|H^2|=\aleph_0$, while the rigid-graph family does not depend on that external theorem.
Editorial extensions
If this is right
- Conjecture 1.1 is false: disjoint subsets smaller than the group need not be separable.
- Every separating set $X$ lies inside $W_G(A,B)$; hence finiteness of $W$ is a certificate of non-separability.
- If $|W_G(A,B)| > \max(|A|,|B|,\aleph_0)$, a countable separating set always exists.
- The sufficient condition cannot be weakened: there are examples with $|W|=\aleph_0$ and no separating set.
- The cardinalities $|A|$ and $|B|$ alone cannot decide separability; the conjugation structure of $A$ is the decisive feature.
Reading between the lines
- A sharper criterion could be obtained by measuring not just the cardinality of $W_G(A,B)$ but the extent to which the conjugation action moves $A$ off $B$; the infinite-$W$ examples suggest such a refinement.
- The rigid-graph construction may transfer to countable permutation groups, which would bear directly on the paper's open question of whether a counterexample with $|G|=\aleph_0$ exists.
- For finite initial segments of the rigid graph on $\mathbb{N}$, one could search directly for an infinite symmetric $X$ with $XAX\cap B=\emptyset$; finding one would refute the paper's claim, while failing to find one would corroborate it.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper addresses Problem 17.102 of the Kourovka Notebook, which conjectured that for any infinite group G and disjoint A,B⊂G with |A|,|B|<|G|, there is an infinite symmetric subset X⊂G with e∈X and XAX∩B=∅. The author defines the witness set W_G(A,B) and proves that any such X must be contained in W (Lemma 3.1), so finiteness of W precludes separability, and that if |W|>λ=max(|A|,|B|,ℵ_0), then a countable witness exists (Lemma 3.2). Two counterexample families are then given: one using Newelski's torsion-free group H with |H|=2^{ℵ_0} and |H^2|=ℵ_0, taking G=H×C_2 and A a singleton; and another using G=Sym(N) with A,B the transpositions corresponding to a rigid graph on N. In both families W is finite. A third and fourth construction (an S_3-valued function group and a semidirect product V⋊U) have infinite W but still admit no witness, showing the strict inequality in Lemma 3.2 cannot be weakened to |W|≥λ.
Significance. The paper gives a negative answer to an open problem from the Kourovka Notebook, which is a valuable contribution to infinite group theory and subset combinatorics. The witness-set formalism is natural and yields a useful necessary/sufficient pair of conditions. A notable strength is that the second counterexample family is fully self-contained in ZFC, so the main refutation does not depend on the external Newelski theorem; the first family is an elegant application of it. The proofs are direct and checkable, and the further examples in Section 6 clarify the exact boundary of the sufficient condition.
minor comments (6)
- [3, Lemma 3.2] The proof states that the total number of forbidden elements is 'less than λ', but the preceding bound gives 'at most λ'; since |W|>λ, the conclusion follows after replacing 'less than' by 'at most'.
- [6.1, Proposition 6.1(2)] The displayed formula for |B| appears to contain a misprint: the term written as 2^{ℵ0} should be ℵ0 (or 2·ℵ0), because both parts of B are countable; as printed, the equality with ℵ0 is contradictory.
- [5, Theorem 5.2] The set B is defined using R^c, which includes diagonal pairs; please specify that B consists of transpositions (u v) with u≠v and u R^c v, to avoid including the identity.
- [2] The labels 'Conjecture 2.1' and 'Conjecture 2.2' are slightly misleading, since 2.1 is disproved and 2.2 is proved in the paper; consider renaming them 'Statement' or 'Question'.
- [1, Theorem 1.1] The word 'conversely' is not accurate; the second assertion is a separate necessary condition rather than the converse of the first.
- [6.1] The phrase 'at some coordinate and e at all others' should be 'at exactly one coordinate and e at all others' for clarity.
Circularity Check
No circularity: Conjecture 1.1 is refuted by a self-contained rigid-graph construction (Section 5), with external Newelski input used only in the independent Section 4 family.
full rationale
The paper's derivation chain is non-circular: every load-bearing object is defined independently of the target statement, and the counterexamples are constructed and verified directly rather than fitted. W_G(A,B) (Definition 3.1) is defined purely by the conjugation products uAu, uAu^{-1}, u^{-1}Au, u^{-1}Au^{-1} avoiding B; Lemma 3.1 (X⊆W for any admissible X) is a direct consequence of that definition, and Lemma 3.2 builds a countable witness set from the hypothesis |W|>λ by excluding at most λ elements (the printed 'less than λ' is a typo for 'at most λ'—a presentation issue, not a circular one). Theorem 1.1 merely packages the two lemmas. The Section 4 counterexample starts from Newelski's external theorem (|H|=2^{aleph_0}, |H^2|=aleph_0, H torsion-free), defines G=H×C_2, A={a}, B={(s,c): s∈H^2\{e}}, and shows every element of any admissible X has first coordinate e, forcing X⊆{e,a}; the conclusion is derived from the definitions and torsion-freeness, not assumed. The Section 5 counterexample is fully self-contained and independent of Newelski: the graph on N with edges {n,m} for 0<n<m≤2n has degree floor(3n/2) strictly increasing, so Aut(Graph)={id}; then for G=Sym(N), A=transpositions on R and B=transpositions on R^c, any g≠id conjugates some edge transposition into a non-edge transposition, so no infinite symmetric X can satisfy XAX∩B=∅, and W={id}. The items labelled Conjecture 2.1 and Conjecture 2.2 are consequences of the constructions (2.1 refuted by Newelski's group, 2.2 proved by the rigid graph), not premises. The unusual insertion 'Using the Tencent Hy3 model, guided by the preceding information' (Section 6) is in-scope provenance: the Section 6 examples directly verify W=E and W=V×{1} and force X={e}, so no step there reduces to its own input. Newelski's theorem is third-party, published, machine-independent input, and is in any case not needed for the Section 5 refutation of Conjecture 1.1. No circular step can be quoted; the honest verdict is no significant circularity, with the noted typos and external dependence being correctness and fragility concerns only.
Assumptions & free parameters
assumptions (4)
- standard math ZFC set theory with the Axiom of Choice
- domain assumption Newelski's theorem: existence of a torsion-free group H with |H|=2^{aleph_0} and |H^2|=aleph_0
- standard math Basic cardinal arithmetic: |Sym(N)|=2^{aleph_0} and |N x N|=aleph_0
- standard math Graph automorphisms preserve vertex degrees
Cite this review
Pith. "Pith review of Counterexamples to Problem 17.102 of the Kourovka Notebook: Negation and Discussion of Separability Conditions in Infinite Groups." pith.science (2026). https://pith.science/paper/IDMVPWHW
@misc{pith2026260800504,
author = {Pith},
title = {Pith review of: Counterexamples to Problem 17.102 of the Kourovka Notebook: Negation and Discussion of Separability Conditions in Infinite Groups},
year = {2026},
howpublished = {\url{https://pith.science/paper/IDMVPWHW}},
note = {Machine review of arXiv:2608.00504}
}
abstract
This paper gives a negative answer to Problem 17.102 of The Kourovka Notebook: there exist an infinite group $G$ and disjoint subsets $A, B \subset G$ satisfying $|A|, |B| < |G|$, yet $A$ and $B$ are not separable in $G$. We introduce the witness set $W_G(A, B)$ and prove a necessary condition and a sufficient condition for separability. Using the torsion-free group constructed by Newelski and rigid binary relations, we obtain two kinds of counterexamples in which $W_G(A, B)$ is finite. Furthermore, we construct examples in which $W_G(A, B)$ is infinite yet separation remains impossible, showing that the sufficient condition cannot be weakened.
Reference graph
Works this paper leans on
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Reviewed August 6, 2026 · model on record in the stance chip above.
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