{"id":"33f4b463-3f98-4d7a-a937-39fdea880ee1","arxiv_id":"2608.00504","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Disjoint subsets of cardinality smaller than an infinite group need not be separable, refuting Problem 17.102.","lead":"This mathematics paper disproves an open conjecture from the Kourovka Notebook by constructing infinite groups whose small subsets cannot be separated. It introduces a new witness-set tool and provides two independent families of counterexamples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Section 5 rigid-relation counterexample is self-contained and correct, so the central refutation of Conjecture 1.1 stands despite typos.","rationale":"The reader's verdict is CONDITIONAL, primarily because of fixable presentation issues and reliance on Newelski's theorem in the first family. I agree that Section 5 provides an independent and correct refutation, so the central claim is secure. The reader's weakest_assumption pointed to Newelski's theorem, but since the second family is self-contained, that dependence is not load-bearing for the main conclusion. I also verified the key computations in Sections 5 and 6: the degree-distinct graph is rigid, the transposition sets satisfy the cardinality requirements, and the contradiction arguments for the nonexistence of an infinite witness set are valid. The only genuine flaws are typos in cardinal arithmetic and inequality wording, which do not alter the mathematical content. Therefore no change to the reader's verdict is warranted.","tokens_in":8102,"tokens_out":22043,"duration_ms":247265,"concrete_test":"Recompute |B| in Proposition 6.1(2) directly from its definition: the first part contributes exactly two elements per coordinate (value (13) or (23) at one coordinate), and the second part is contained in the countable set E, so the union is countable and the printed cardinality is a typo. Independently, compute the degree sequence of the Section 5 graph for n=0..20 and check it is strictly increasing, confirming Aut(Ω,R)={id} and the self-contained counterexample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I examined the two counterexample families and the witness-set lemmas. Section 5 gives a self-contained ZFC counterexample to Conjecture 1.1: the graph on N with E={{n,m}: 0<n<m≤2n} has strictly increasing degrees, hence Aut(Ω,R)={id}; the transposition sets A and B have cardinality aleph0 < 2^{aleph0}=|Sym(N)|, and for any g≠id there is an edge transposition a with g a g^{-1} a non-edge transposition in B, so no infinite symmetric X can satisfy XAX∩B=∅. This argument is independent of Newelski's theorem, so the dependence of Section 4 on external results is not load-bearing for the central claim. The remaining defects are presentation typos: the printed '2^{aleph_0}' in Proposition 6.1(2) should be '2·aleph_0' (the set B is countable), and 'less than λ' in Lemma 3.2 should be 'at most λ'. Neither affects the main theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8375,"tokens_out":20963,"duration_ms":209568,"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.","major_comments":[],"minor_comments":[{"comment":"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'.","section":"3, Lemma 3.2"},{"comment":"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.","section":"6.1, Proposition 6.1(2)"},{"comment":"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.","section":"5, Theorem 5.2"},{"comment":"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'.","section":"2"},{"comment":"The word 'conversely' is not accurate; the second assertion is a separate necessary condition rather than the converse of the first.","section":"1, Theorem 1.1"},{"comment":"The phrase 'at some coordinate and e at all others' should be 'at exactly one coordinate and e at all others' for clarity.","section":"6.1"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound. The central refutation is supported by a self-contained construction in Section 5, so correctness does not hinge on the external Newelski theorem. The remaining issues are typographical and should be fixed in revision; the paper is short but the result is substantial enough for publication in a group theory journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers the first counterexamples to Kourovka Notebook Problem 17.102, and the core mathematics is sound. The witness set W is a genuinely useful tool: Lemma 3.1 is a simple necessary condition, Lemma 3.2 gives a sufficient condition, and the two counterexample families show the boundary is real. The Section 5 construction is the highlight—a graph on N with strictly increasing degrees, so its automorphism group is trivial, and the transposition sets A and B give W={id}. That argument is entirely self-contained, in ZFC, and does not depend on any external theorem. The Section 4 example using Newelski's group also works, and it is nice that the paper shows both a finite-W and an infinite-W obstruction, so the sufficient condition cannot be weakened to |W|≥λ.\n\nThe soft spots are real but minor, and all in presentation. Proposition 6.1(2) prints |B|=2^{aleph_0}, but B is countable; the intended value is aleph_0, and the conclusion is unchanged. Lemma 3.2 says the number of forbidden elements is 'less than λ' when the bound is actually 'at most λ'—again harmless. The disclosure that part of Section 6 was produced with the Tencent Hy3 model is unusual; it does not affect the mathematics, but an editor may want the authors to clarify their own contribution. The first counterexample family depends on Newelski's theorem, but the second family removes that dependence, so the refutation of Conjecture 1.1 does not rest on an external result.\n\nI checked the reader's flagged points and they hold: W={e,a} in Section 4, W={id} in Section 5, and the Section 6 examples genuinely force X={e}. The cardinality typos are exactly typos. The central claim—that |A|,|B|<|G| does not imply separability—is correct and well-supported.\n\nThis paper deserves a serious referee. It resolves an open problem from a standard source, the method is reusable, and the main proof is transparent. I would send it to review with a request that the author fix the typos and clarify the AI-assisted writing statement. I would cite it if I worked in subset combinatorics of groups, and I would bring it to a reading group as a compact example of how a naive cardinality conjecture fails in a concrete way.","headline":"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.","tokens_in":8846,"tokens_out":1328,"would_cite":true,"duration_ms":15429,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F05","20B07","05C25","03E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["infinite groups","separability","witness sets","torsion-free groups","square sets","rigid binary relations","Kourovka Notebook","subset combinatorics"],"falsifier":"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.","tokens_in":7826,"feed_emoji":"","tokens_out":8434,"duration_ms":92408,"temperature":0.7,"pith_summary":"This paper refutes a conjecture from the Kourovka Notebook that asserted: in every infinite group, any two disjoint subsets each of size smaller than the group can be separated by an infinite symmetric subset containing the identity. The author defines a witness set $W_G(A,B)$ of group elements whose conjugates of $A$ still avoid $B$, and proves that any separating set must lie inside $W_G(A,B)$. A sufficient condition is also proved: when $|W_G(A,B)| > \\max(|A|,|B|,\\aleph_0)$, a countable separating set exists. The counterexamples make the witness set finite, or make every possible witness force the candidate set to collapse to $\\{e\\}$, so that the conjecture fails. The paper further shows that the condition \"$W$ infinite\" is not enough to guarantee separability.","feed_headline":"Counterexample refutes separability conjecture for infinite groups","feed_subtitle":"Even when both subsets are smaller than the group, no infinite symmetric set separates them in the constructed examples.","key_machinery":"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 $W=\\{id\\}$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The Kourovka Notebook, source of Problem 17.102 and Conjecture 1.1 that the paper refutes.","marker":"[1]"},{"why":"Supplies the torsion-free group with $|H|=2^{\\aleph_0}$ and $|H^2|=\\aleph_0$ used in the first counterexample family.","marker":"[2]"}],"fun_headline_variants":["Counterexample shows infinite separability conjecture false","Witness set, not size, decides separability in infinite groups","Infinite group counterexample: small subsets not separable","Separability fails even when subsets are smaller than group","New counterexample refutes Kourovka separability problem"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Counterexample shows infinite separability conjecture false","Witness set, not size, decides separability in infinite groups","Infinite group counterexample: small subsets not separable","Separability fails even when subsets are smaller than group","New counterexample refutes Kourovka separability problem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000155,"raw_usage":{"total_tokens":1211,"prompt_tokens":936,"completion_tokens":275,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":552,"completion_tokens_details":{"reasoning_tokens":197}},"tokens_in":552,"tokens_out":275,"duration_ms":3071,"temperature":1.0,"reasoning_tokens":197,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T04:13:27.204490+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Newelski,On the number of squares in a group, Proc","cited_arxiv_id":null,"evidence_quote":"Supplies the torsion-free group with $|H|=2^{\\aleph_0}$ and $|H^2|=\\aleph_0$ used in the first counterexample family."}],"review_version":2}