{"id":"e05d7a73-1565-4a0c-909b-8fc4e24f3839","arxiv_id":"2607.21410","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Selection structures admit Fan–Radon and Volovikov-type Borsuk–Ulam theorems whose conclusions are governed by Radon and Tverberg partitions, yielding new ham-sandwich and necklace-splitting results.","lead":"This paper proves new Borsuk–Ulam-type theorems for 'selection structures', a broad framework that includes matroids and chessboard complexes. The results generalize recent Radon-style strengthenings of Fan's theorem and Volovikov's Tverberg-type theorem, with applications to fair partition problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"D-admissibility of the chessboard and matroid selection structures is deferred to [Sob26]; the central Theorem 3 depends on it, so the advertised Corollaries 3–5 are conditional.","rationale":"The reader's weakest-assumption identification is exactly the load-bearing concern. The proof of Theorem 3 is internally coherent: the skeleton-induction works if the connectivity clause of Definition 2 holds, the equivariance and limiting arguments are standard, and the application proofs faithfully instantiate the selection structures. The only serious gap is that the two families of selection structures that make the paper novel — matroidal and chessboard — are asserted to be d-admissible solely by reference to the author's unpublished preprint [Sob26]. No proof is given in this manuscript, and the connectivity requirement is specifically for parallel expansions with arbitrary nonempty finite fibers, which is stronger than ordinary connectivity of K[U]. If [Sob26] does not establish this, then the central theorem does not imply the advertised Corollaries 3–5. This is not a fatal internal error; it is a conditional support problem. The reader's CONDITIONAL verdict is therefore appropriate, and no adjustment is needed. The proposed concrete test — verifying the exact statement in [Sob26] and, as a spot-check, computing homology of a small chessboard parallel expansion — would settle whether the concern lands.","tokens_in":16627,"tokens_out":25670,"duration_ms":237378,"concrete_test":"Check the companion preprint [Sob26]: does it prove that for the matroid structure (Def. 3) and the chessboard structure (Def. 5), K(U;D) is (d−1)-connected for every U∈L and every indexed family of nonempty finite fibers D_w — not merely for unit fibers? Independently re-derive the chessboard case for the minimal stair a=b=4, d=2 with fibers of size 2 by computing H_1 of K_st,2(U;D); if H_1≠0, Definition 5 is false. If the proof in [Sob26] is valid for arbitrary fibers, the concern is resolved and the verdict stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 2 makes Theorem 3 conditional on two properties: (i) for every U∈L and every assignment of nonempty finite fibers D_w, the parallel expansion K(U;D) is (d−1)-connected; (ii) facets of K[U] are facets of K. The facet condition is automatic for the d-skeleta used in the applications. The connectivity condition is the load-bearing part, and for the two advertised non-matroidal/matroidal families it is not proved here: Definition 3 (matroid) says 'the author has shown ... [Sob26]' and Definition 5 (chessboard) says 'The existence ... was also established in [Sob26].' The proof of Theorem 3 invokes this connectivity at the skeleton-induction step ('Since Γ(τ) is (d−1)-connected, the map ψ can be extended over τ'). If [Sob26] proves only connectivity of K[U] with unit fibers, or requires fibers of size one, then Γ(τ) need not be (d−1)-connected for the arbitrary finite fibers D_w(τ) that arise from the cover, and the extension step fails. In that case Corollaries 3–5 (doubly-colorful, matroid, chessboard) would not follow from Theorem 3. This is a deferred-support issue, not an internal inconsistency: the simplex and sparse specializations are elementary.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces selection structures — pairs (K,L) consisting of a simplicial complex K on a finite ground set W and an upward-closed family L of subsets — and proves two Borsuk–Ulam-type theorems governed by them. Theorem 3 (selection-structure Fan–Radon) states that, for a d-admissible selection structure, whenever closed sets A^w_i⊆S^d (w∈W, i∈[m]) satisfy a cover condition on complements of L and a disjointness condition on edges of K, one obtains disjoint nonempty P+,P−⊆W×[m] with intersecting convex hulls of associated point sets, a common intersection x∈∩_{P+}A^w_i∩∩_{P−}(−A^w_i), π1(P+∪P−) a facet of K, and π2(P+)∩π2(P−)=∅. Theorem 5 is a prime-power analogue governed by Volovikov's theorem, with r-part Tverberg partitions and q-wise label disjointness. The paper then specializes Theorem 3 to simplex, sparse, doubly-colorful, matroid, and chessboard settings (Corollaries 1–5), gives a Fan-type Volovikov corollary (Corollary 6), and derives ham-sandwich and necklace-splitting applications (Theorems 7–8). The proofs use the (d−1)-connectivity of parallel expansions K(U;D), skeleton-wise equivariant extension, Borsuk–Ulam/Volovikov, and the Sarkaria–Bárány–Onn tensoring trick.","tokens_in":16957,"tokens_out":49178,"duration_ms":434905,"significance":"If correct, the paper gives a unifying framework that recovers Frick and Wellner's recent Fan–Radon theorems and extends them to matroidal, sparse, and non-matroidal (chessboard) settings, together with Tverberg/Volovikov and fair-division variants. The proof of Theorem 3 is clean: it isolates the needed hypothesis (connectivity of parallel expansions), uses a standard odd-map/Borsuk–Ulam argument, and the Volovikov proof invokes the Sarkaria–Bárány–Onn tensoring correctly. The paper is also transparent about the exact sense in which it recovers the Frick–Wellner conclusion (Remark 1). The principal weakness is verification debt: the d-admissibility of the matroid and chessboard selection structures is deferred to the author's unpublished preprint [Sob26], and the arbitrary-fiber connectivity there is exactly the load-bearing input to the main theorems.","major_comments":[{"comment":"Definition 2 requires that for every U∈L and every family D of nonempty finite fibers, the parallel expansion K(U;D) be (d−1)-connected. This is load-bearing: in the proofs of Theorem 3 (§2) and Theorem 5 (§4), the induction over skeleta extends ψ across a simplex τ because Γ(τ)=K(U(τ),D(τ)) is (d−1)- (resp. (n−1)-) connected, where the fibers D_w(τ) are arbitrary nonempty finite sets generated by the cover. For the matroid structure (Definition 3) and the chessboard structure (Definition 5), d-admissibility is asserted by citing the author's preprint [Sob26]. If [Sob26] proves connectivity only for unit fibers or for a different class of complexes, the extension step fails and Corollaries 4–5 are unsupported. Please include the admissibility proofs (or exact statements with matching hypotheses) in the manuscript, or explicitly mark Corollaries 4–5 as conditional on [Sob26]. The doubly-c","section":"§3, Definitions 3 and 5; Corollaries 4–5"},{"comment":"The compactness step at the end of the proof of Theorem 3 ('the finiteness of W and the compactness of S^d imply by a standard argument that the same holds for some point x in the limit') is too terse for a step that delivers the conclusion x∈∩_{(w,i)∈P+}A^w_i ∩ ∩_{(w,i)∈P−}(−A^w_i). The argument should pass to a subsequence on which the pairs (P_+^α,P_−^α) are constant, note that the diameter of the containing face τ_α tends to 0 so all witnessing vertices converge to a common point x, and invoke closedness of the A^w_i. The convex-hull intersection and the facet property are preserved along the constant subsequence. The same abbreviated argument appears in the proof of Theorem 5 and should be expanded there as well.","section":"§2 (proof of Theorem 3); §4 (proof of Theorem 5)"}],"minor_comments":[{"comment":"The running title contains a typo: 'GENERALIZA TIONS'. Please also check the accent formatting of the author name in the abstract.","section":"Title/header"},{"comment":"Notation abuses: 'we have {u,w}∈Γ(τ), which in turn implies {u,w}∈K' should be phrased in terms of the support of the face η; also the parenthetical 'D(τ) is taken as a multiset, as we want one fiber for each w∈W' is unclear, since Definition 2 states connectivity for families indexed by W while D(τ) is indexed by U(τ).","section":"§2 (proof of Theorem 3)"},{"comment":"The d-admissibility of the sparse structure (K = d-skeleton of 2^[n], L = subsets of size ≥ d+1) is asserted without verification. Add a sentence: for |U|=d+1, K(U;D) is the join of d+1 nonempty fibers and hence (d−1)-connected; for |U|>d+1 it is the (d+1)-partite skeleton of the join, whose (d−1)-connectivity follows by the standard nerve/join-skeleton argument.","section":"§3, Corollary 2"},{"comment":"The final sentence 'The conclusion of Theorem 3 corresponds exactly to the desired conclusion' should be spelled out: define f(w) as the unique index i with (w,i)∈P±, and use x∈∩_{P+}A^w_i∩∩_{P−}(−A^w_i) to obtain H=∂H+∈∩_{w∈B}G_w.","section":"§5, Theorem 7"},{"comment":"In the merging argument for the diagonal case, state explicitly that after removing a duplicate same-sign label, the projection π1(P+∪P−) remains a face of K (as a subset of the original face), so the modified conclusion is legitimate.","section":"§3, Corollary 1"},{"comment":"The definition of a rainbow sequence could note that the consecutive blocks overlap in one element, matching the Bukh–Loh–Nivasch usage.","section":"§4, Corollary 6"},{"comment":"The paper depends crucially on the unpublished preprint [Sob26]; the relevant statements (with full hypotheses) should be quoted or the dependence clearly flagged in the introduction.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central theorem and its proof are sound in my reading; the decision hinges on the deferred d-admissibility of the matroid and chessboard selection structures. The author cites his own unpublished preprint [Sob26] for exactly the property (arbitrary-fiber connectivity) that Theorem 3 and Theorem 5 consume. I would ask the editor to verify that [Sob26] is available (e.g., on arXiv) and states the connectivity result for arbitrary nonempty finite fibers, or require the author to prove it in this paper. The rest of the manuscript is in good shape; the limit-argument sketch should also be expanded. This is a verification gap rather than an internal error."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper gives a clean selection-structure generalization of Fan's theorem (Theorem 3) and a Volovikov-style analogue (Theorem 5), and it recovers the usual colorful, sparse, matroid, and chessboard variants. The proofs are standard once the definition is granted, but the definition is doing real work. The main caveat is that the matroid and chessboard selection structures are not proved admissible here; that is deferred to the author's companion [Sob26]. If [Sob26] only proves connectivity for unit fibers, the paper's Corollaries 3–5 don't follow. That is a load-bearing conditional, not a nitpick.\n\nWhat is good: Theorem 3 is genuinely new, and the proof is coherent. The skeleton-induction step uses exactly the (d−1)-connectivity of K(U;D), the δ-almost-disjointness argument for sign-disjointness is sound, and the facet condition is applied correctly. The simplex and sparse specializations are elementary and self-contained. The tensoring argument in Theorem 5 is correctly invoked, and the ham-sandwich application (Theorem 7) is clean. The necklace-splitting application is a natural use of the Volovikov version. The paper is clearly written, and the citations to Fan, Volovikov, BLN, and the author's own prior work are appropriate.\n\nSoft spots: the deferred admissibility. Definition 2 requires connectivity for all nonempty finite fibers, and the D_w(τ) built in the proof are arbitrary finite subsets of E. If the companion only proves the unit-fiber version, the extension step in Theorem 3 fails. The paper does not state exactly which theorem from [Sob26] it relies on. The limit argument in both main proofs is sketched but standard. Corollary 6 compresses the Bukh–Loh–Nivasch use; I think it works, but it should be checked carefully. The self-citation is heavy but not abusive—the definition genuinely comes from there.\n\nBottom line: for anyone working in topological combinatorics or discrete geometry, this is worth reading and discussing. It deserves a serious referee. The referee should have access to [Sob26] and verify the admissibility claims, or ask the author to include those proofs. My own verdict is conditional: I would accept if the companion holds up, and I would not desk-reject.","headline":"Solid selection-structure Fan/Volovikov package with a real caveat: matroid and chessboard admissibility are deferred to the companion [Sob26], and the advertised corollaries hang on that.","tokens_in":17433,"tokens_out":13035,"would_cite":true,"duration_ms":117406,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A35","55M20","55M35","05E45"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that d-admissible selection structures yield Borsuk–Ulam-type covering theorems with Radon or Tverberg partitions, unifying a family of colorful and sparse results.","keywords":["Borsuk–Ulam theorem","selection structures","Radon partitions","Tverberg partitions","Volovikov theorem","matroids","chessboard complexes","mass partitions"],"falsifier":"Compute the connectivity of the parallel-expansion complex K(U;D) for a specific U∈L and small fibers, for example the chessboard selection structure with d=2, |X|=|Y|=4, U a (4,4,3)-stair, and |D_w|=2 for all w; if the complex is not (d−1)-connected, the chessboard structure is not d-admissible and the chessboard Fan–Radon corollary would be false as stated.","tokens_in":16507,"feed_emoji":"🧩","tokens_out":11753,"duration_ms":112578,"temperature":0.7,"pith_summary":"This paper proves that a broad class of combinatorial constraints, called selection structures, can be imposed on Borsuk–Ulam-type theorems without losing their force. The main result, Theorem 3, says that if a finite set W carries a d-admissible selection structure (a simplicial complex K plus an upward-closed family L), and closed sets A^w_i on the d-sphere satisfy a covering condition for every sufficiently large subset and a disjointness condition along the edges of K, then a Radon-type partition must exist: two disjoint subsets P+, P− of W×[m] whose point sets have intersecting convex hulls, whose labels are disjoint, and whose first-coordinate projection is a facet of K. A prime-power analogue, Theorem 5, replaces the antipodal action by the action of (Z_p)^a and yields Tverberg partitions. These theorems specialize to recover and extend recent colorful Fan theorems, as well as sparse, matroidal, and chessboard-complex versions, and they yield selection-structure forms of the ham sandwich and necklace splitting theorems. In this way the paper identifies the combinatorial shape—d-admissible selection structures—that underlies a large family of covering results.","feed_headline":"Selection structures unify Borsuk–Ulam theorems","feed_subtitle":"A single setup covers matroids, chessboards, and fair-division theorems from ham sandwiches to necklaces.","key_machinery":"The key mechanism is the d-admissible selection structure: a pair (K,L) where K is a simplicial complex on a finite set W and L is an upward-closed family of subsets, satisfying that for every U∈L and every indexed family of nonempty finite fibers D_w, the parallel-expansion complex K(U;D)—the join over the vertices of each face of K[U] of the corresponding fibers—is (d−1)-connected, and that facets of K[U] are also facets of K. This connectivity lets the proof extend a partial equivariant map over each simplex of a fine triangulation of S^d, producing a continuous odd map ψ:S^d→|K(W;E)|. Composing ψ with the linear map that sends (w,±i) to (±x^w_i,±1) yields an odd map S^d→R^d; by the Borsu","core_discovery":"The paper's central claim is that d-admissible selection structures are the right general setting for a whole family of Borsuk–Ulam-type covering theorems. Theorem 3 states: if W is a finite set and Σ=(K,L) is d-admissible, then for any assignment of m points in R^{d-1} to each w∈W and any closed subsets A^w_i⊂S^d, the assumptions (i) the sphere is covered by the sets A^w_i∪(−A^w_i) for every W' whose complement is not in L, and (ii) A^u_i∩(−A^w_i)=∅ whenever {u,w}∈K, force the existence of disjoint nonempty P+,P−⊆W×[m] such that the convex hulls of the two selected point sets intersect, some x∈S^d lies in both ⋂_{P+}A^w_i and ⋂_{P−}(−A^w_i), the first-coordinate projection of P+∪P− is a fac","pith_inferences":["If the cited admissibility of the chessboard and matroid structures indeed holds, the same extendability scheme should apply to other well-connected, facet-inheriting complexes—matching complexes and shifted complexes are natural next candidates for new colorful Borsuk–Ulam theorems.","The proof's reliance on fine triangulations and a limit argument hints that the conclusion may survive for families of open or merely measurable sets with a suitable approximation, which could broaden the ham-sandwich and necklace-splitting applications beyond closed sets.","Because the necklace theorem is restricted to r a prime power, testing the selection-structure conclusion for composite r (e.g., r=6) would reveal whether the prime-power restriction is essential or merely an artifact of Volovikov's theorem."],"forward_implications":["The simplex specialization—W=[d+1], K the full simplex, L={W}—recovers the recent Radon-type strengthening of Fan's theorem, with P+ and P− giving intersecting convex hulls and intersecting set-intersections.","The sparse version, with K the d-skeleton of a simplex, yields a Radon partition of exactly d+1 pairs whenever the sphere is covered by any n−d of the n labeled families.","The matroid version, with K the d-skeleton of a matroid independence complex, yields the partition on a facet of rank d+1 whenever the covering holds for all subsets whose complement has rank at most d.","The chessboard version, for X×Y with ν(|X|,|Y|)≥d+1, yields a placement of d+1 non-attacking rooks, a genuinely non-matroidal specialization.","The Volovikov-type theorem, specialized to W=[n+1] and K=2^W, gives a Fan-type covering version of Volovikov's theorem whose Tverberg partitions are rainbow, and that reduces to the classical Fan theorem when r=2."],"fun_headline_variants":["Selection structures generalize Borsuk-Ulam theorems","Borsuk-Ulam, now with selection structures","One framework for Borsuk-Ulam and fair division","Selection structures unify Borsuk-Ulam and fair division","Borsuk-Ulam via selection structures, from matroids to necklaces"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the selection structures used in the applications—especially the matroid and chessboard ones—really are d-admissible; for those two examples the required connectivity of the parallel-expansion complexes is cited from another paper, not proved in this one.","fun_headline_variants_meta":{"raw":{"variants":["Selection structures generalize Borsuk-Ulam theorems","Borsuk-Ulam, now with selection structures","One framework for Borsuk-Ulam and fair division","Selection structures unify Borsuk-Ulam and fair division","Borsuk-Ulam via selection structures, from matroids to necklaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001263,"raw_usage":{"total_tokens":4988,"prompt_tokens":705,"completion_tokens":4283,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":449,"completion_tokens_details":{"reasoning_tokens":4200}},"tokens_in":449,"tokens_out":4283,"duration_ms":30316,"temperature":1.0,"reasoning_tokens":4200,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:31:47.147171+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the connectivity of the parallel-expansion complex K(U;D) for a specific U∈L and small fibers, for example the chessboard selection structure with d=2, |X|=|Y|=4, U a (4,4,3)-stair, and |D_w|=2 for all w; if the complex is not (d−1)-connected, the chessboard structure is not d-admissible and the chessboard Fan–Radon corollary would be false as stated.","supporting_citations":[],"review_version":1}