{"id":"6dac147a-4a0f-475d-802d-9831cbb292bd","arxiv_id":"2411.17573","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A zero-dimensional metrizable CDH space X is constructed such that X^2 is not CDH, because X^2 has exactly continuum many pairwise non-equivalent countable dense subsets.","lead":"A topology paper builds a space X that is 'countable dense homogeneous' (its countable dense subsets can be shuffled onto each other by homeomorphisms), while the product space X^2 loses this property. The construction works in ordinary set theory and answers a question that had been open in the field.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The counting argument in Theorem 7 defines F_n as both countable and homeomorphic to 2^ω; no such set exists, and neither reading yields a countable dense D with the claimed invariant.","rationale":"The reader's weakest_assumption (Y^2 contains no copy of 2^ω) is a genuine but minor gap, likely repairable by a compactness argument. The reader also noted the F_n contradiction in the rationale, and the stress-test identifies that as the load-bearing issue. Without F_n ≈ 2^ω, the entire dichotomy between Q0 and Q1 collapses: because every countable dense set is countable, the stated Q0 property is vacuous, so Corollary 5 cannot be doing the claimed work. With F_n ≈ 2^ω, D is uncountable, so X^2 has not been shown to have c types of countable dense subsets. Thus the central claim is unsupported by the current manuscript. This is not an unstated assumption but an explicit internal inconsistency, so the verdict should move from CONDITIONAL to REJECT. A future version that replaces this step with a coherent construction may become reviewable, but the submitted text does not contain such a construction.","tokens_in":3742,"tokens_out":16235,"duration_ms":151477,"concrete_test":"Formalize the proposition: there exists a countable dense set Q1 and, for each n ∈ ω, a set F_n ⊆ Q1 with F_n homeomorphic to 2^ω. Prove or disprove this in ZFC. Since any subset of a countable set is countable and the Cantor set is uncountable, no such F_n exists, which settles the inconsistency. Equivalently, attempt to replace \"countable\" with \"closed\" and check whether the subsequent contradiction h(F_n) ⊂ Q0 survives: it does not, because the Q0 property only forbids subsets homeomorphic to 2^ω, not closures.","verdict_should_be":"REJECT","load_bearing_attack":"In Theorem 7, the proof of \"exactly c types\" hinges on the step: \"For n ∈ ω let F_n ⊂ U_n be countable such that F_n ≈ 2^ω. Let Q1 = ⋃ F_n.\" This is impossible: a countable set cannot be homeomorphic to the Cantor set. If \"countable\" is taken literally, then every h(F_n) is countable, so the alleged contradiction with \"for no countable subset E ⊂ Q0 we have E ≈ 2^ω\" vanishes; moreover, that quoted property is true of every countable Q0, so Corollary 5 supplies no nontrivial invariant. If \"≈ 2^ω\" is taken literally, then Q1 is a countable union of uncountable sets and hence uncountable, so D = Q0 ∪ Q1 is not a countable dense subset, and the proof does not produce c types. The two requirements are irreconcilable. This is not a missing justification but an inconsistent object at the center of the counting argument, so the ZFC construction of X^2 with exactly c types is not established by the submitted text.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a ZFC construction of a zero-dimensional metrizable CDH space X with the property that X^2 has exactly c types of countable dense subsets, and consistently this space can be co-analytic. The construction takes a CDH λ-set Y of cardinality aleph_1, forms X = Y ⊕ 2^ω, and attempts to show that every homeomorphism of X^2 preserves the clopen summand 2^ω × Y, after which a family {D_r} of countable dense subsets of 2^ω × Y indexed by r ∈ (0,1) is built by adjoining pairwise nonhomeomorphic countable nowhere dense sets C_r to a common countable core D. The paper claims to answer an open question of Medini and to give a consistent co-analytic example. The proof is forward and uses external theorems ([1], [3], [4], [10]) in a non-circular way.","tokens_in":3959,"tokens_out":17724,"duration_ms":161524,"significance":"If the main theorem were established, it would answer a question of Medini and complement the earlier consistent construction under MA(σ-centered) with a ZFC example. The use of λ-sets and the strategy of identifying a clopen summand preserved by all homeomorphisms are natural and potentially useful. The proof is not circular: it relies on previously known results rather than on its own conclusions. However, the central counting argument contains an impossible object, and several supporting claims about the λ-set Y are not justified. As written, the main theorem is not established.","major_comments":[{"comment":"The definition of F_n is internally inconsistent: the text says 'For n ∈ ω let F_n ⊂ U_n be countable such that F_n ≈ 2^ω.' No countable set is homeomorphic to the Cantor set. If 'countable' is taken literally, then h(F_n) is countable for every homeomorphism h, so it cannot contradict Corollary 5's statement 'for no D ⊂ C we have D ≈ 2^ω'; the alleged contradiction disappears. If '≈ 2^ω' is taken literally, then Q1 = ⋃ F_n is uncountable, so D = Q0 ∪ Q1 is not a countable dense subset, and the construction does not produce countable dense sets D_r of the desired kind. This is not a missing justification but an impossible requirement at the center of the c-types counting argument.","section":"§3, Theorem 7"},{"comment":"The step 'By Lemma 3, there is a collection {C_r; r ∈ (0,1)} of countable pairwise nonhomeomorphic nowhere dense subsets of {s} × Y' is not justified. Lemma 3 gives pairwise nonhomeomorphic nowhere dense subsets of Q, but the paper does not prove that these spaces can be realized as countable nowhere dense subsets of {s} × Y. This would require, for instance, an embedding of Q (or of each relevant countable space) into {s} × Y and a verification that the embedded copies are nowhere dense in {s} × Y. Without this transfer, the final contradiction h(C_p) = C_r is unsupported.","section":"§3, Theorem 7"},{"comment":"The claim that π2(h(2^ω) ∩ (2^ω × Y)) 'is a clopen subset of Y therefore it is a crowded space' assumes without proof that Y has no isolated points. A CDH λ-set can have isolated points (for example, a space homeomorphic to 2^ω with one isolated point added is CDH), and a clopen subset of such a space need not be crowded. Since the conclusion that this set is homeomorphic to 2^ω is used to prove h(2^ω) = 2^ω, this missing hypothesis is load-bearing for the invariance argument.","section":"§3, Theorem 7"}],"minor_comments":[{"comment":"The abstract states that X^2 'has exactly c countable dense subsets', whereas the body and the intended theorem concern 'exactly c many types of countable dense subsets'. These are different statements and the abstract should be corrected.","section":"Abstract"},{"comment":"The paper repeatedly uses the fact that Y^2 contains no copy of 2^ω, but only the remark that no λ-set contains a copy of 2^ω is stated. The product statement is plausibly true for λ-sets, but it should be proved or explicitly cited.","section":"§2, Remark after Definition 1"},{"comment":"The phrase 'any clopen subset of 2^ω × Y contains the Cantor set' should be qualified to nonempty clopen subsets; otherwise the statement is false for the empty set.","section":"§3, Theorem 7"},{"comment":"The decomposition 'X^2 ≈ Y^2 ⊕ 2^ω ⊕ 2^ω × Y' compresses the full expansion (Y ⊕ 2^ω)^2; it would help to note explicitly that the two cross-product summands are identified under 2^ω ⊕ 2^ω ≈ 2^ω and that 2^ω × 2^ω ≈ 2^ω.","section":"§3, Theorem 7"},{"comment":"In the statement of Theorem 6, the notation 'ω L 1' is garbled; it should presumably read '(ω_1)^L'.","section":"§3, Theorem 6"}],"recommendation":"reject","confidential_remarks":"The central counting argument in Theorem 7 is built on an object that cannot exist. This is not a local gap that a revision can repair while preserving the proof strategy; the contradiction between 'countable' and 'homeomorphic to 2^ω' is the load-bearing mechanism for producing many types. The intended fix mentioned by the readers would require a substantially different invariant. I therefore recommend rejection, despite the interesting question and the natural use of λ-sets."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper has the right target and a plausible route, but the submitted proof does not establish the result. The central counting argument in Theorem 7 relies on sets F_n ⊂ U_n that are simultaneously countable and homeomorphic to the Cantor set. No such set exists. If you read 'countable' literally, every h(F_n) is countable and the contradiction with Q0 disappears; if you read '≈ 2^ω' literally, Q1 is uncountable and D is not countable. Either way, the proof of exactly c types collapses. This is not a missing justification; it is an inconsistent object at the center of the argument.\n\nThat said, the paper is not a waste of time. The construction X = Y ⊕ 2^ω, where Y is a CDH λ-set of cardinality ℵ1, is a sensible ZFC strategy that avoids the MA(σ-centered) assumption in Medini's partial result. The use of external theorems ([1], [3], [4], [10]) is non-circular, and the paper is honest about what it relies on. The descriptive-set-theoretic part (co-analytic under MA+¬CH+ω1=(ω1)^L) follows cleanly from Miller's theorem if Y is Π^1_1.\n\nBeyond the F_n contradiction, the proof has other soft spots that are minor by comparison but should be addressed in any revision: the fact that Y^2 contains no copy of 2^ω for a λ-set Y is used without proof, and the projection argument in the invariance step assumes that a clopen subset of Y is crowded, which is not automatic for λ-sets. These are gaps, not contradictions, and are probably fixable.\n\nWho should read this? Anyone working on countable dense homogeneity or definable CDH spaces. The open problem is real and the strategy is worth pursuing. But the current text cannot be published as is. I would send it back for major revision with a clear request to fix the counting argument, not desk-reject it. A corrected version deserves a serious referee.","headline":"Plausible ZFC construction for an open CDH problem, but the submitted proof has a fatal contradiction in the counting argument.","tokens_in":4440,"tokens_out":2963,"would_cite":false,"duration_ms":26801,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["54G20","54H05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs, in ZFC alone, a zero-dimensional metrizable CDH space $X$ whose square $X^2$ is not CDH, settling an open question.","keywords":["countable dense homogeneous","lambda-set","zero-dimensional","metrizable","co-analytic","product homogeneity","Cantor set","countable dense subsets"],"falsifier":"Check whether $Y\\times Y$ contains a copy of $2^\\omega$ for the $Y$ supplied by Theorem 2; if it does, the paper's invariance argument fails. Equally, if some clopen subset of $Y$ is not crowded, the projection step no longer yields a copy of $2^\\omega$, and the conclusion that $h$ preserves $2^\\omega\\times Y$ has no basis.","tokens_in":3553,"feed_emoji":"♾️","tokens_out":9174,"duration_ms":84029,"temperature":0.7,"pith_summary":"The paper proves that countable dense homogeneity (CDH) is not preserved by taking squares, even among zero-dimensional metrizable spaces. It builds such a space $X$ in ZFC, with the stronger property that $X^2$ has exactly $\\mathfrak{c}$ many types of countable dense subsets. This settles an open question from the literature and improves on a previous construction that required an additional set-theoretic axiom. Consistently, the space can be taken to be co-analytic. The proof is elementary, using only $\\lambda$-sets and standard facts about Cantor-set neighborhoods.","feed_headline":"A CDH space with a non-CDH square exists in ZFC","feed_subtitle":"The square has exactly continuum many types of countable dense subsets—the maximum possible—and consistently the space is co-analytic.","key_machinery":"The load-bearing object is the disjoint sum $X=Y\\oplus 2^\\omega$ with $Y$ a CDH $\\lambda$-set; a $\\lambda$-set is a subspace of $2^\\omega$ in which every countable subset is relatively $G_\\delta$, and no $\\lambda$-set contains a copy of $2^\\omega$. The invariance step is carried by the paper's Corollary 5, derived from the proof of a known product theorem: in a product of a space that contains $2^\\omega$ with one that does not, some countable dense set has no countable subset homeomorphic to $2^\\omega$. This corollary certifies that the component $Q_0$ of the constructed dense set cannot absorb a Cantor set, while Lemma 3 supplies $\\mathfrak{c}$ pairwise nonhomeomorphic nowhere dense subsets of $\\mathbb{Q}$ used as tags $C_r$.","core_discovery":"Set $X=Y\\oplus 2^\\omega$, where $Y$ is a CDH $\\lambda$-set of cardinality $\\aleph_1$; Theorem 2 supplies such a $Y$ for any cardinal up to $\\mathfrak{b}$. The paper proves that every homeomorphism of $X^2$ must preserve the clopen summand $2^\\omega\\times Y$: a clopen piece of that summand contains a copy of $2^\\omega$, while $Y^2$ does not, and the projection of any Cantor set that tried to cross into $2^\\omega\\times Y$ would give a clopen crowded subset of $Y$, hence a copy of $2^\\omega$ inside $Y$. Inside $2^\\omega\\times Y$, the proof builds $\\mathfrak{c}$ countable dense subsets $D_r$ by taking one fixed dense set $D=Q_0\\cup Q_1$ and adjoining pairwise nonhomeomorphic nowhere dense sets $C_r\\subseteq\\{s\\}\\times Y$. Any homeomorphism moving $D_p$ to $D_r$ would have to send a point of $\\{s\\}\\times Y$ into a Cantor-set half, forcing a copy of $2^\\omega$ into the $\\lambda$-set side; the resulting contradiction shows the types are distinct, so $X^2$ is not CDH.","pith_inferences":["A natural extension, not stated in the paper, is that the same construction may make $X^n$ non-CDH for every $n\\ge 2$ by applying the clopen-summand argument to $2^\\omega\\times Y^{n-1}$; the needed check is whether $Y^k$ remains free of copies of $2^\\omega$ for all $k$.","The counting mechanism is modular: it only needs one factor with a Cantor-set neighborhood and one Cantor-free factor with enough nonhomeomorphic nowhere dense subsets, so the 'exactly $\\mathfrak{c}$ types' conclusion may hold for a broader class of CDH spaces than the particular sum $Y\\oplus 2^\\omega$.","Because the construction is in ZFC while the descriptive upgrade is only consistent, the boundary between ZFC examples and projective examples appears to be genuinely set-theoretic; this is suggested by the paper's reliance on a consistency result for co-analyticity."],"forward_implications":["The square of the constructed space has exactly $\\mathfrak{c}$ types of countable dense subsets, so the open question is answered negatively: CDH is not preserved by squares even in ZFC.","The construction also settles the $\\kappa=\\mathfrak{c}$ case of the question asking which cardinals can occur as the number of types for such a space.","The existence of the space is unconditional, unlike the earlier example that used an additional set-theoretic axiom.","Consistently, the example is a co-analytic subspace of $2^\\omega$, which is best possible in the sense that Borel zero-dimensional CDH spaces are classified and have CDH squares."],"supporting_citations":[{"why":"Supplies Theorem 2: for every cardinal $\\kappa\\le\\mathfrak{b}$ there is a CDH $\\lambda$-set of size $\\kappa$, which provides the factor $Y$.","marker":"[4]"},{"why":"Supplies Corollary 5, the product obstruction used to choose a countable dense $Q_0\\subset A\\times Y$ with no countable Cantor-set subset.","marker":"[3]"},{"why":"Supplies Lemma 3, the $\\mathfrak{c}$ pairwise nonhomeomorphic nowhere dense subsets of $\\mathbb{Q}$ used as the tags $C_r$.","marker":"[1]"},{"why":"Supplies Theorem 6, the consistency result that every $\\aleph_1$-sized subset of $2^\\omega$ is co-analytic under the stated axioms, used for the co-analytic version of the space.","marker":"[10]"}],"fun_headline_variants":["CDH space with a non-CDH square exists in ZFC","ZFC construction: CDH space, non-CDH square","Medini's question answered: CDH space with non-CDH square","Square of a CDH space can have continuum many dense types","CDH property not inherited by square, even in ZFC"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the unproved fact that $Y^2$ contains no copy of $2^\\omega$ (and that every clopen subset of $Y$ is crowded); if a CDH $\\lambda$-set of cardinality $\\aleph_1$ failed either condition, the step forcing every homeomorphism to preserve $2^\\omega\\times Y$ would collapse.","fun_headline_variants_meta":{"raw":{"variants":["CDH space with a non-CDH square exists in ZFC","ZFC construction: CDH space, non-CDH square","Medini's question answered: CDH space with non-CDH square","Square of a CDH space can have continuum many dense types","CDH property not inherited by square, even in ZFC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001164,"raw_usage":{"total_tokens":4786,"prompt_tokens":882,"completion_tokens":3904,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":3815}},"tokens_in":498,"tokens_out":3904,"duration_ms":29158,"temperature":1.0,"reasoning_tokens":3815,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:59:57.571599+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether $Y\\times Y$ contains a copy of $2^\\omega$ for the $Y$ supplied by Theorem 2; if it does, the paper's invariance argument fails. Equally, if some clopen subset of $Y$ is not crowded, the projection step no longer yields a copy of $2^\\omega$, and the conclusion that $h$ preserves $2^\\omega\\times Y$ has no basis.","supporting_citations":[{"cited_title":"Hern´ andez-Guti´ errez, M","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 2: for every cardinal $\\kappa\\le\\mathfrak{b}$ there is a CDH $\\lambda$-set of size $\\kappa$, which provides the factor $Y$."},{"cited_title":"Hern´ andez-Guti´ errez","cited_arxiv_id":null,"evidence_quote":"Supplies Corollary 5, the product obstruction used to choose a countable dense $Q_0\\subset A\\times Y$ with no countable Cantor-set subset."},{"cited_title":"Brian, J","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3, the $\\mathfrak{c}$ pairwise nonhomeomorphic nowhere dense subsets of $\\mathbb{Q}$ used as the tags $C_r$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 6, the consistency result that every $\\aleph_1$-sized subset of $2^\\omega$ is co-analytic under the stated axioms, used for the co-analytic version of the space."}],"review_version":1}