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Follow-up (2026-09-27): more stale "open" labels, found by syncing with sources that maintain status data. Again, this is only meant to save others duplicated effort.

1. Erdős problems. All 632 EP-* entries are labelled open in v1.6.0. As of today, erdosproblems.com lists 90 of them as resolved (we fetched each page; many resolutions are recent, several by AI systems, and most are Lean-verified). Grouped by the site's status:

• Proved, Lean-verified (28): EP-38, EP-123, EP-126, EP-152, EP-258, EP-281, EP-283, EP-330, EP-351, EP-358, EP-369, EP-457, EP-469, EP-557, EP-571, EP-610, EP-750, EP-793, EP-825, EP-865, EP-987, EP-997, EP-1014, EP-1022, EP-1051, EP-1071, EP-1096, EP-1129
• Proved (9): EP-380, EP-591, EP-652, EP-851, EP-863, EP-986, EP-1021, EP-1105, EP-1130
• Disproved, Lean-verified (14): EP-1, EP-43, EP-74, EP-90, EP-125, EP-146, EP-180, EP-193, EP-533, EP-846, EP-847, EP-884, EP-990, EP-1092
• Disproved (8): EP-92, EP-543, EP-574, EP-575, EP-705, EP-869, EP-960, EP-992
• Solved otherwise (e.g. determined/answered), Lean-verified (12): EP-42, EP-119, EP-183, EP-190, EP-202, EP-318, EP-619, EP-650, EP-694, EP-696, EP-741, EP-888
• Solved otherwise (e.g. determined/answered) (19): EP-320, EP-321, EP-346, EP-387, EP-421, EP-477, EP-603, EP-625, EP-633, EP-690, EP-730, EP-783, EP-858, EP-896, EP-920, EP-948, EP-1005, EP-1089, EP-1091

The site's machine-readable status file (teorth/erdosproblems, data/problems.yaml) may be the easiest way to keep these entries in sync.

2. Ben Green's 100 open problems. The current version of the list (updated December 2025) marks these as solved. Note that the dataset's GREEN-xxx numbers differ from the numbering in Green's list.

3. OWR-1452-012 (Zhao's Vanishing Conjecture for homogeneous quartics). Zhao proved that this conjecture, over all n, is equivalent to the Jacobian conjecture over all n (Trans. AMS 359 (2007), arXiv:math/0409534). The July 2026 Jacobian counterexample (see SMA-016 above) therefore refutes it for some n; we have not identified the smallest such n.

— Alper Ferudun

Follow-up 2 (2026-09-27): Kourovka Notebook, issue 21 (KOU-21.*). The arXiv version of the notebook updated today (arXiv:1401.0300v46) marks the following issue-21 problems as solved (asterisk), while v1.6.0 still labels them open:

(The other nine starred issue-21 problems, 21.8, 21.12, 21.14, 21.15, 21.18, 21.24, 21.43, 21.58, 21.150, are already marked solved in the dataset.)

— Alper Ferudun

Follow-up 3 (2026-09-27): KOU-21.68 — new counterexample (our own result, unrefereed). Kourovka Notebook Problem 21.68 (M. Kida) conjectures that every finite semi-abelian group is monomial. This is false. There is a semi-abelian group of order 768 = 2^8·3 with a non-monomial irreducible character of degree 8:

• Ĝ = B ⋊ W, where W = E ⋊ A₄ is the index-two subgroup of C₂ ≀ A₄, T ≅ SL(2,3) ≤ W is the binary tetrahedral group acting on the eight quaternion units, and B is the augmentation (even-weight) submodule of the permutation module F₂[W/T].
• General reduction (Clifford theory): if a linear character of the abelian normal subgroup N has stabiliser T in W and T has a non-monomial irreducible character, then N ⋊ W is not an M-group.
• Checked by exact computation, including every subgroup of index 8 and the full character table of Ĝ (exactly the three degree-8 irreducible characters are non-monomial). Scripts are in the source archive.

Preprint: https://doi.org/10.5281/zenodo.23000305 · paper page: https://eulersolve.org/papers/kou-21-68/

Suggested label: solved (answered negatively). It has not been peer-reviewed yet, so independent checks are welcome. Minimality of the order is not claimed; Kida's Magma search covered all orders up to 240.

— Alper Ferudun

Follow-up 4 (2026-09-27): internal status inconsistency. For 188 problem numbers, the literature-triage block inside the record itself ("Literature review (checked 2026-08-17)", Status: solved, Classification: SOLVED-IN-LITERATURE) disagrees with the status field, which still says open (same in v1.6.0 and v1.7.0). 94 of them are already covered in the comments above; the remaining 94 are listed below by source so they can be reconciled in one pass. We have not re-verified each triage conclusion ourselves, and some of them refute a literal wording rather than the intended question, so they are pointers, not claims.

• HIL (1): HIL-009
• SMA (1): SMA-004
• NT (5): NT-008, NT-023, NT-035, NT-037, NT-086
• GREEN (3): GREEN-061, GREEN-064, GREEN-100
• GEO (3): GEO-004, GEO-006, GEO-028
• GEOM (1): GEOM-026
• GT (1): GT-008
• GRAPH (4): GRAPH-006, GRAPH-008, GRAPH-046, GRAPH-049
• TOP (1): TOP-003
• ALG (2): ALG-033, ALG-036
• COMB (2): COMB-012, COMB-004
• HL (1): HL-F
• GUY (3): GUY-A8a, GUY-A11, GUY-A15
• KP (6): KP-1.51, KP-3.14, KP-4.37, KP-4.125, KP-5.9, KP-5.15
• OPG (51): OPG-23298, OPG-50149, OPG-37185, OPG-426, OPG-1797, OPG-37167, OPG-37181, OPG-37230, OPG-692, OPG-37086, OPG-610, OPG-37341, OPG-34908, OPG-37081, OPG-37089, OPG-37218, OPG-37316, OPG-37364, OPG-57613, OPG-59952, OPG-59997, OPG-824, OPG-46606, OPG-47285, OPG-59911, OPG-2242, OPG-59994, OPG-616, OPG-36939, OPG-52200, OPG-47031, OPG-47643, OPG-37305, OPG-47646, OPG-677, OPG-690, OPG-691, OPG-735, OPG-177, OPG-157, OPG-732, OPG-760, OPG-2379, OPG-37444, OPG-37863, OPG-37402, OPG-655, OPG-1783, OPG-37245, OPG-37295, OPG-57401
• EP (9): EP-129, EP-520, EP-524, EP-545, EP-550, EP-612, EP-638, EP-654, EP-996 — note that erdosproblems.com still lists all nine as open (EP-550 as "open (Lean)"), so these triage conclusions deserve a second look before relabelling.

A JSON list (problem number, status field, triage status) is available on request.

— Alper Ferudun

Hello, thank you for these detailed additions! They are now integrated.

Follow-up 5 (2026-09-28): two new results, one problem already solved in the literature, and corrections to Oberwolfach and Kourovka records (stale statuses, transcription errors, merged records, one misattached note). Each item was checked against the current version (commit 2ea030b). Where a page is given, it was also checked in the report itself; literature was checked against arXiv and Crossref. Suggested changes are in bold.

New results (unrefereed preprints with full proofs, verification scripts and independent referee checks)

• OWR-12861-021 → solved (no). Heinig's Question 2 (OWR 01/2014, p. 82) has a negative answer for every odd n ≥ 7: K_{(n+1)/2,(n−1)/2} with a perfect matching (or a matching plus one P₃) inside the larger side has minimum degree ⌈n/2⌉ but no spanning copy of the near-square, under both readings of "periphery"; at n = 9, K_{4,4,1} is a counterexample under every reading. For n = 7 the host is Heinig's own graph X (arXiv:1112.5101, Def. 28). The cycle-space Question 1 is not affected (Hou–Yin). Paper: doi:10.5281/zenodo.23004012 (page).
• KOU-21.76 → solved. The existence question was first answered by Ya. N. Nuzhin, Sib. Math. J. 67 (2026) 840–845, Corollary 1 (examples in every characteristic, over F(y,z)); the notebook (v46) does not record this yet. Our note gives explicit examples with short proofs, including one-variable examples over F₃(t) (which also satisfy the hypotheses of 19.48), and shows that in characteristic 3 such nets exist over K iff K is not algebraic over F₃. Paper: doi:10.5281/zenodo.23004034 (page).

Stale statuses

• OWR-12330-013 → solved (yes). King answers his own Question 1 on the next page: "The answer to this is yes" (OWR 02/2013, p. 105). K_k with k pendant vertices at every vertex has χ_f = k > 3k/4 + 1 for k ≥ 5.
• OWR-1319-022 → solved (no). Two word metrics on H₃(ℤ)×ℤ have ratio → 1 but unbounded difference: Breuillard, Groups Geom. Dyn. (2014) (arXiv:0704.0095), and Breuillard–Le Donne, PNAS 110 (2013), §5, who cite this report.
• OWR-5158-016 → solved (yes). By Deroin–Hurtado (arXiv:2008.10687, Thm 1.3), irreducible lattices in finite-centre semisimple groups of real rank ≥ 2 are not left-orderable, so any cocompact arithmetic lattice in SL(3,ℝ) works. See also Witte Morris's exposition (2026).
• OWR-14213-006 → solved. The statement is the unit-interval (Hessenberg) form of the Stanley–Stembridge conjecture, proved by Hikita, J. Amer. Math. Soc. (2026) (arXiv:2410.12758).
• OWR-1386-016 → solved, if "threshold" has its usual coarse meaning (the constant b is then immaterial). This is the two-graph Kohayakawa–Kreuter conjecture: the 1-statement is due to Mousset–Nenadov–Samotij (CPC 2020), and Christoph–Martinsson–Steiner–Wigderson (Proc. LMS 130, 2025) completed the proof. A sharp threshold at b·n^(−1/m₂) would be a different question; we could not re-read the wording in OWR 48/2006.
• OWR-14299911-005 → solved (disproved). The source itself reports Chalopin–Chepoi's counterexample to the "MSO decidable ⇔ grid-free" conjecture (ACM TOCL 20 (2019)). Please also drop the second sentence: the special-cube-complex result concerns Thiagarajan's other conjecture, and the counterexample itself comes from a virtually special complex.
• OWR-17294-009 → at least partially solved. Part (1), Byott's question, is answered no by Di Matteo–Ferrara–Trombetti, arXiv:2607.22795 (July 2026 preprint). The record bundles Vendramin's Problems 1, 3 and 5 (OWR 51/2019, p. 3222); splitting it would help.
• KOU-21.115 → solved (yes). Sambale, arXiv:2609.09052 (September 2026 preprint), proves |G∖U| ≥ |G|/2ⁿ whenever n left cosets have union U ≠ G, and right cosets are left cosets of conjugates. The notebook (v46) has not starred it yet.
• OWR-4213-002 → solved. The statement is the Feit–Thompson theorem (Pacific J. Math. 13 (1963)). Also, "nontrivial" simple groups should read "non-abelian".
• OWR-1265-024 → not an open problem. The source poses it as a puzzle and gives the answer C(a+b,a) − C(a+b,a−1), which counts standard (b,a)-tableaux (G. James, OWR 15/2006, pp. 965–966).
• Pointer only, OWR-11568-005. Lutowski's Theorem 2 (Publ. Math. Debrecen 99 (2021)) says that the holonomy of a non-torus Kähler flat manifold has at least two distinct irreducible constituents. Since h^{1,1}(A/G) = dim End_G(T₀A), this seems to force b₂ ≥ 2 for free torus quotients of dimension ≥ 2. Worth a check.

Transcription errors

• OWR-12330-010, -011, -012: the ceilings in the source are missing (OWR 02/2013, p. 103, Conj. 3; p. 104, Conj. 4–6). As written, C₅ violates all of them: the right-hand sides are 5/2 or 3 − ε, while χ(C₅) = 3.
• OWR-1189-007: Kohl's Conjecture 2 reads ⌊3d(1−1/n)⌋ + 1, not ⌈·⌉ (OWR 7/2006, p. 416). As written it is false for d = 1 and n ≥ 4, because χ^{1,1}_ℓ(P_n) = ch(P_n²) = 3.
• OWR-5154-009: the source puts the profinite closure on the product [g₁]^F⋯[g_k]^F, not on N(g₁,…,g_k) (OWR 26/2011, p. 1452, Conj. 2). As written, the answer is trivially no.
• OWR-12007-008: with "r, s ∈ ℝ" (the same wording as OWR 35/2012, p. 2173) the question is false for w = a, since R(a,r,s) = r. The source's own results are for rational r, s, so this should read r, s ∈ ℚ.

Merged records

• OWR-4791-001 merges two conjectures that the organisers' introduction (OWR 1/2011, p. 6) reports as proved: Simonovits–Sós (Ellis–Filmus–Friedgut) and Sumner (for large n, Kühn–Mycroft–Osthus). Splitting it would help.
• OWR-9790352-039, OWR-10252930-031, OWR-9790352-036: the statement is already clean, but original_statement still contains a neighbouring item, respectively:
  • the Plummer–Zha conjecture, proved during the workshop (OWR 1/2022, p. 71; Chudnovsky–Seymour, JGT 103 (2023));
  • a weighted Turán conjecture, proved by Bradač (arXiv:2205.08923);
  • Narayanan's permanent inequality (OWR 1/2022, pp. 69–70), which is open and has no record of its own.

Misattached note

• OWR-12861-020: the literature note (Hou–Yin, arXiv:2503.15950) is about Heinig's Question 1 (OWR 01/2014, p. 81), which has no record of its own. It does not concern the Diestel or Friedgut questions in this record, and the record's "partially solved" label rests only on that note.

Correction to Follow-up 2. arXiv:1401.0300v46 (Kourovka Notebook No. 21) was posted on 1 September 2026, not on 27 September as I wrote there; the list of starred problems is unaffected.

— Alper Ferudun

Follow-up 6 (2026-09-28): three new results, and corrections to Oberwolfach combinatorics records (stale statuses, questions answered in their own source, transcription errors, duplicates, merged records, garbled extracts and misattached notes). Each item was checked against the current version (commit 2ea030b). Where a page is given, it was also checked in the report itself; literature was checked against arXiv and Crossref. Suggested changes are in bold.

New results (unrefereed preprints with full proofs, verification scripts and independent referee checks)

• OWR-16164-019 → solved (no). Sportiello's conjecture B_λ ≥ A_λ (OWR 23/2018, pp. 1455–1457) fails for the band λ = {(x,y) ∈ [7]² : −3 ≤ y−x ≤ 4}, where A_λ = 536 and B_λ = 515. Every digitally convex shape of side ≤ 6 satisfies the inequality; among the 5,693,968 shapes of side 7, only this band and its transpose violate it. (The record attaches the red and blue crosses to the opposite colours; that swap is a bijection on colourings and does not change B_λ.) Paper: doi:10.5281/zenodo.23006715 (page).

• OWR-1782-009 and OWR-1386-013 (the same conjecture) → false as stated. Both ask that minimum codegree ⌊(n−k+3)/2⌋ force a tight Hamiltonian cycle for all n: for n ≥ k+1 ≥ 4 in OWR 01/2008 (p. 46), and with no range of n in OWR 48/2006 (p. 2928). This all-n form is Conjecture 1.1 of Rödl–Ruciński–Szemerédi, Adv. Math. 227 (2011), who attribute it to Katona–Kierstead. It fails for (k,n) = (3,7), (3,9), (4,8), (5,9) and (5,10). The smallest counterexample is an apex joined to all 15 pairs of a 6-set W, plus the ten faces of the hemi-icosahedron on W. Its pair-degrees are 3 and 5. But in a cyclic order (apex, w₁, …, w₆) the disjoint windows w₁w₂w₃ and w₄w₅w₆ would both have to be faces, and no two faces are disjoint. The (5,10) example has codegree 4 = (n−k+3)/2, so the version without the floor fails too. The large-n statement is not affected: RRS proved it for k = 3, and Letzter–Lang–Ranganathan–Sanhueza-Matamala have announced it for all k ≥ 3, as reported in arXiv:2609.08613. Suggested status: "false as stated (small counterexamples); the large-n form is proved for k = 3 and announced for all k". Also:

  • OWR-1386-013 should say "tight" Hamiltonian cycle, as its source defines it; under a Berge reading these hypergraphs do have Hamiltonian cycles;
  • mark the two records as duplicates;
  • the literature DOIs on OWR-1782-009 concern other thresholds (10.1016/j.jcta.2015.01.004: Hamilton ℓ-cycles with ℓ < k/2; 10.1112/jlms.12561: minimum d-degree conditions). RRS 2011 and arXiv:2609.08613 are the relevant references.

  Paper: doi:10.5281/zenodo.23006718 (page).

• OWR-14299577-018 → solved (yes, both questions). The problem was proposed by C. Bernert and N. Arala Santos, in the problem session compiled by T. F. Bloom (OWR 51/2025, p. 2756), so proposed_by can be filled in. Let A ⊂ {−n,…,n}∖{0} contain exactly one of k and −k for every k ≤ n. Then at most two elements of [1,n] are missing from A − A. This is sharp: {1,…,a} ∪ {−(a+1),…,−n} misses exactly a and a+1 whenever n/2 ≤ a ≤ n−1. For every B ⊆ [1,n], the number of pairs (a,b) ∈ A² with a − b ∈ B is at least ⌊(|B|−1)²/4⌋, and this is attained for every |B| ≤ ⌊2n/3⌋+1. So for |B| ≥ εn the count is at least (1/4 − o(1))ε²n². The key step is that the representation counts r(d) of any set S of differences of one parity satisfy Σ_{d∈S} r(d) ≥ C(|S|,2), by a double count of the positions where the sign pattern repeats at distance d. The record's original_statement is garbled: it drops the set-up sentence and splices in two sentences from the preceding problem (Assing). Paper: doi:10.5281/zenodo.23006720 (page).

Stale statuses

• OWR-16160-016 → solved as stated (classical). Valette asks whether the congruence subgroups of G ⊂ SL_N(ℤ) depend on the embedding (OWR 19/2018, p. 1151). In general they do. For example, F₂ embeds as ⟨(1 2; 0 1), (1 0; 2 1)⟩ and as ⟨(1 3; 0 1), (1 0; 3 1)⟩. The level-2 subgroup of the second embedding contains no congruence subgroup of the first, and the level-3 subgroup of the first contains none of the second, so the two congruence topologies are incomparable. Classical instances go back to Serre: SLₙ(ℤ) has congruence subgroups whose images under the adjoint map are not congruence subgroups (Lubotzky–Venkataramana, Algebra Number Theory 13 (2019), Prop. 2.1). For the groups of the talk, Z² ⋊_A Z, the answer is the opposite, and the same holds for every solvable G. Solvable groups have the congruence subgroup property (Chahal, Nagoya Math. J. 79 (1980)), and solvable subgroups of GLₙ(ℤ) are polycyclic, so every embedding induces the profinite topology (LV, §1.1). A scope note would help.
• OWR-1265-006 → solved (yes). Olsson's containment question (OWR 15/2006, p. 913) is the Olsson–Stanton conjecture. Vandehey proved it (arXiv:0809.2134, 2008), and Fayers gave another proof (JCTA 118 (2011)).
• OWR-1536-011 → solved (no, both parts). Knox (arXiv:1212.3345) gives a hypergraph on which Breaker, moving second, wins Maker–Breaker, yet Chooser wins Chooser–Picker. Knox notes that the Picker–Chooser part is equivalent to it; the equivalence comes from the transversal-hypergraph duality stated in the source (OWR 20/2007, p. 1095). So the transversal hypergraph of his example refutes the Picker–Chooser part too.
• OWR-4425-020 (duplicate -021) → solved (yes). Shallit's Conjecture 48 (OWR 37/2010, p. 2236) is Theorem 18 of Cassaigne–Currie–Schaeffer–Shallit, J. ACM 61 (2014), for exactly this morphism.
• OWR-2090-021 → partially solved. Adiprasito–Björner (arXiv:1401.7301, Thm 2.1) prove the Mikhalkin–Ziegler conjecture from this problem session:
  • for a generic weight and t ≤ min{0, total weight}, the proper flats of weight > t form a homotopy Cohen–Macaulay poset, hence an (r−3)-connected one;
  • the "non-negative" version follows by a small perturbation;
  • they credit rank 3 to Pinchasi–Ziegler (personal communication, 2008);
  • shellability is still open (their Open Problem 3.2).
• OWR-12481-012 → solved (no). When Friedgut re-posed it for fixed t and large n, the report recorded: "This turns out to be false; a counterexample was found by Gábor Tardos" (OWR 22/2016, p. 1217). The question as posed here already fails for n = 4, t = 2. Pair σ with σ∘(1 2) if σ maps {1,2} onto {1,2} or {3,4}, and with σ∘(3 4) otherwise. The resulting 12 two-cosets refine none of the 8 partitions of S₄ into 1-cosets.
• OWR-16633-025 → solved (no), over every field. We found no written answer, but a 4-element example settles it: T₀ = F², T₁ = ⟨e₁⟩, T₂ = ⟨e₂⟩, T₃ = ⟨e₁+e₂⟩. Then f(0) = 2, f(1) = f(2) = f(3) = 1, and f(S) = 2 for every other nonempty S. Suppose f = Σ c_j r_j with c_j > 0 and matroid ranks r_j on {0,1,2,3}; the r_j need not even be representable.
  • Every matroid has r(0i) ≥ r(0) and r(ij) ≤ r(i) + r(j). Since f attains equality in both, so does every r_j.
  • So in each r_j, the elements 1, 2 and 3 lie in the closure of 0, which has rank ≤ 1, and at most one of them is a non-loop.
  • Hence r_j(123) = r_j(1) + r_j(2) + r_j(3) for every j, and summing gives f(123) = 3. But f(123) = 2.
  • Over GF(2), 2·r(U₂,₄) is another example. Dougherty–Freiling–Zeger (arXiv:0910.0284, §4) represent it over every field. Its only possible matroid summands are copies of U₂,₄, which is not binary.

Answered in the source itself

• OWR-4791-006 → solved. Fox–Lee–Sudakov prove both conjectures in the abstract itself (OWR 01/2011, pp. 11–12: Thm 2, f(m) = ⌊√(4m+1)⌋ − 1, and Thm 4). The paper is Israel J. Math. 191 (2012). The record's own verification note already says so.
• OWR-16931-001 → solved. The record asks only about sufficiently large n. For that case the source says "We answer this affirmatively for all sufficiently large n" (OWR 19/2019, p. 1158). This is Glock–Joos–Kim–Kühn–Osthus, JEMS 23 (2021).
• OWR-14299518-002 → solved. Alon answers both questions in the abstract itself (OWR 42/2025, pp. 2249–2250): (1) no (Thm 3), (2) yes (Thm 4). These are Erdős problems #664 (disproved) and #732 (proved). The note's "structural characterization" is not part of this record.
• OWR-12872-010 → solved, and one formula needs fixing. Stanley (with F. Liu) proves both of Elkies' conjectures in the same abstract (OWR 12/2014, pp. 696–697); the paper is Ramanujan J. 36 (2015). For n = 2m the count of maximum families is 2^((m−1)(m−2))·(2^m − 1), not ·(2m − 1). A brute-force count gives 28 for n = 6 and 960 for n = 8.

Transcription errors and literal readings

• OWR-14298158-004: Claesson's conjecture is monotonicity in the length n for fixed k: |Av_n^k(1324)| ≤ |Av_{n+1}^k(1324)| (OWR 6/2024, p. 284; Claesson–Jelínek–Steingrímsson, JCTA 119 (2012)). The same correction applies to the Av(1324, 231) part. As written (k → k+1), the statement is false for every n, e.g. |Av_{2,1}| = 1 > 0 = |Av_{2,2}|.
• OWR-17135-030, -031, -032: the source prints M_ii = Σ_{S∋i}|S| (OWR 39/2019, p. 2464). This is a typo for Σ_{S∋i} X_S = deg(i).
  • With that diagonal, det M = |X|·|Y|·τ(G)/∏_{y∈Y} deg y. So Conjecture 5 becomes equivalent to Ehrenborg's Conjecture 4, as the source says.
  • As printed, Conjecture 5 is false. Take X = [3], hang 13 leaves on each vertex of X, and add one vertex joined to all three. Then det M = 850 > 512 = det(diag M). It also fails if the sum runs only over the sets S that occur.
  • With the corrected diagonal, Conjecture 5 is Ehrenborg's conjecture, proved by Ho (arXiv:2603.17997; -030 already cites it).
  • The refinement in -031/-032 is false as literally stated, already at n = 3. There det(diag M) − det M is homogeneous of degree 3 with coefficient −1/4 on X₁₂X₁₃X₂₃. No such polynomial equals Σc_μx^μ + Σc_μν(x^μ − x^ν)² with all c ≥ 0, because the square terms would contribute a nonzero part of even degree. If multipliers x^ρ(x^μ − x^ν)² were intended, the record should say so.
• OWR-14299089-007: as written, the question is trivial: (a_n b_n)² ≥ a_{n−1}a_{n+1}b_{n−1}b_{n+1} for nonnegative sequences. The source asks Brändén–Ferroni–Jochemko's Question 6.1 (Trans. AMS 2026, arXiv:2408.12386). Write Σ p(n)xⁿ = W(p)/(1−x)^(deg p+1). If W(p) and W(q) are log-concave with no internal zeros, is W(pq)?
• OWR-11695867-010: as the source itself says (OWR 57/2022, p. 3274), the case ℓ = 0, Σ f_λ² = n!, is "exactly the Robinson-Schensted-Knuth algorithm". Louf's Open problem 1 is a bijective proof of n!·H_{n,ℓ} = Σ_{λ⊢n} f_λ² C_λ^ℓ for all ℓ. Here H_{n,ℓ} counts ℓ-tuples of transpositions with product 1, and C_λ is the content sum.
• OWR-16164-005: the literal question has a classical answer. Put y_i = 1 − x_i for odd i; each constraint then becomes y_i ≤ y_{i+1} or y_i ≥ y_{i+1}. So the polytope is the order polytope of a fence, a poset whose Hasse diagram is the path 1–2–⋯–n. Its Ehrhart polynomial is Ω(P, t+1) (Stanley, DCG 1 (1986)), so h* is the descent polynomial of its linear extensions (natural labelling). We checked this for all 63 sign sequences with n ≤ 6. Mark solved, or restate if another interpretation was intended.

Duplicates

• Across reports: OWR-12481-012 is the Friedgut half of OWR-12861-020 (OWR 18/2013, p. 1118; OWR 01/2014, p. 81). Splitting OWR-12861-020 would leave there only Diestel's k-block question, which is open.
• New: OWR-14299904-003 repeats -002, which is Weigandt's Conjecture 1 (OWR 2/2026, p. 133); -004 is Conjecture 2.
• Already flagged: the following records are marked as repeats in their own statement_verification, but are still published as open or partially solved. So each of these problems is counted twice. A duplicate status, or unpublishing, would help. The records are OWR-734-010, OWR-1183-007, -012, OWR-4425-019, -021, OWR-4791-016, OWR-4798-022, OWR-11136-011, OWR-14604-015, OWR-16633-022, -024, OWR-16763-026, -028, OWR-17135-032 and OWR-1703876-017.

Merged records

• OWR-1703876-016 (duplicate -017) bundles Problems 6–10 of the OWR 30/2020 problem session (p. 1522), posed by five different people. Splitting would help. Two of its notes need fixing:
  • Problem 10 (Welzl, partial triangulations) is solved (yes) by Kupavskii–Volostnov–Yarovikov, Europ. J. Combin. 108 (2023) (arXiv:2104.05855). -017 lists that arXiv paper under the names Aichholzer–Orden–Schnider.
  • Problem 8 (Steiner, bichromatic triangles) is still open. -016 says a 2026 preprint proves it. In fact Radtke–Keszegh–Lauff (arXiv:2601.20574) prove it only for at most 5 red pseudolines (so for n ≤ 11). In general they prove only that a two-coloured triangle or quadrangle exists.
• OWR-4425-003 bundles Currie's Open problems 2–4 (OWR 37/2010, p. 2206): Restivo–Salemi reachability, the curling-number conjecture and the lexicographically least 5/2-power-free word. Splitting would help.
• OWR-2090-006 bundles Linial's open-ended challenges on Latin squares and the Γ function with one precise conjecture (OWR 44/2008, p. 2496): the maximum rank of a real n×n×n tensor is (1+o(1))n²/2. That conjecture deserves its own record.
• OWR-2090-026: the statement is clean (Problem 9, Barvinok–Samorodnitsky), but original_statement also contains Problem 10 (Welzl, spanning trees versus triangulations). That problem is already OWR-2090-027.

Garbled extracts (all already marked "unrecoverable", but still published as open or partially solved)

• OWR-12861-024: the source itself is ill-posed (OWR 01/2014, p. 83). The matrix has rows indexed by S_n but columns only by {σ : LIS(σ) ≥ n−t}, and it is called both M and A, so its determinant is undefined.
• OWR-9790358-007: the opening of Hakopian's abstract plus its title (OWR 7/2022, p. 405), with no question in it. Its only conjecture, Gasca–Maeztu, is OWR-9790358-001.
• OWR-12697684-002, -003: table-of-contents lines (OWR 1/2023, pp. 9–10). -012: background sentences from Bucić's abstract (p. 42) about the Erdős–Hajnal conjecture, which is -011.
• Also in OWR 1/2023: the "original OWR report" link of OWR-12697684-001, -002, -003, -011 and -012 points to 10.4171/owr/2022/57 (Enumerative Combinatorics). It should be 10.4171/owr/2023/1.
• OWR-14298158-001: a table-of-contents entry (OWR 6/2024, p. 277). -016: the closing remark of an abstract on Bevan's conjecture (p. 299); the conjecture itself is -015.
• OWR-723-001: Beck's four circle-discrepancy questions. The source itself reports them as answered, by Schmidt and by Beck (OWR 13/2004, pp. 678–679). The conjecture that remains is OWR-723-002. Mark solved or remove.

Misattached notes

• OWR-2090-021: the note (parametric assignment, rotation matching) belongs to Rote's Problem 2 of the same session, least-squares matching under rotation (OWR 44/2008, pp. 2546–2547). The relevant literature for this record is Adiprasito–Björner (above).
• OWR-4425-012: the note ("sum-square avoidance", Au–Robertson–Shallit) is about the additive-square problem, Problem 47, which is OWR-4425-018. The record itself is Shallit's Problem 38 on pattern characterisations of α-powers (OWR 37/2010, p. 2231).

— Alper Ferudun

Scoped algorithmic result related to AMR-029-0087: Divisorial Gonality of Banana Trees in Quadratic Time (Alper Ferudun, Mercury Software GmbH; 5-page preprint).

For every connected loopless multigraph whose underlying simple graph is a tree, the paper gives an exact algorithm computing its divisorial gonality and an attaining effective divisor in O(n^2) word operations and O(n^2) space after reading the input. This covers arbitrary branching and binary edge multiplicities, without expanding parallel edges. The proof uses a rooted residue normal form, a rerooting formula, independent child costs conditional on the parent pile, and sliding-window minima.

Scope and prior work: this does NOT resolve the general multigraph bound gon(G) <= floor((g+3)/2) in the frozen dataset record. Beougher et al., Chip-Firing Games on Banana Trees, arXiv:2510.03499v1, already proved that bound for banana trees and gave a cubic algorithm for banana paths. Those results are credited, not claimed as new. The present contribution is the quadratic exact algorithm with a divisor witness for arbitrary banana trees; the original record should remain open on the strength of this note alone.

The reproducibility package includes exact checks on 2,657 instances, 445,497 effective divisors, 13,071 root choices and 65,005 witness-target checks, with an independent reduced-Laplacian oracle and normal/optimized Python runs. These finite checks support, but do not replace, the general proof. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute priority is claimed.

DOI and PDF/source/verification files: https://doi.org/10.5281/zenodo.23467234
Paper page: https://eulersolve.org/papers/amr-029-0087/
Closest located prior result: https://arxiv.org/abs/2510.03499

OWR-12170-005: restore the elementary hypothesis; distinguish a known literal counterexample

The frozen 1.6.0 entry states the depth-two Hilbert-series assertion for every finite-dimensional Nichols algebra. That unrestricted sentence is false, but this does not refute the intended elementary conjecture.

The OWR report, pp. 2901–2902, ends with an unqualified sentence, following a discussion with irreducibility hypotheses. Heckenberger–Lochmann–Vendramin, Conjecture 1.1, p. 2, explicitly says elementary: the degree-one Yetter–Drinfeld module is absolutely irreducible and its support generates the group. Its footnote identifies the October 2012 Oberwolfach workshop.

For the unrestricted reading, take the standard diagonal Cartan-type A3 braiding over C with

Q = ((-1,-1,1), (1,-1,-1), (1,1,-1)),
c(x_i tensor x_j) = q_ij x_j tensor x_i.

The known order-two PBW description (Andruskiewitsch–Angiono, §4.1, pp. 40–41) gives

$$H(t)=(1+t{)}^3(1+t^2{)}^2(1+t^3)={\mathrm{\Phi }}_2(t{)}^4{\mathrm{\Phi }}_4(t{)}^2{\mathrm{\Phi }}_6(t),H(1)=64.H(t)=(1+t)^3(1+t^2)^2(1+t^3)=\backslash Phi\_2(t)^4\backslash Phi\_4(t)^2\backslash Phi\_6(t),\backslash qquad\; H(1)=64.$$

Every product of (a)_t and (b)_(t^2) has v_Phi3 >= v_Phi6: for the first kind, divisibility by 6 implies divisibility by 3; for the second, the two multiplicities are equal. This H has multiplicities 0 and 1, so no such product exists.

One group realization is over (C2)^3, with deg(x_j)=g_j and g_i x_j=q_ij x_j. Each line C x_j is a submodule, so this example is not elementary. It therefore does not answer the qualified conjecture. Exact rational quantum-shuffle calculations independently reproduce the Hilbert coefficients and a zero degree-11 component.

Suggested annotation: restore the elementary scope and distinguish it from the false unqualified wording. This is a known-family reconstruction and source correction, not a new algebra, new paper, priority claim, or resolution of the intended problem. AI-assisted and self-audited; no independent review or maintainer-applied change is claimed.

OPG-58213: the odd-order Odd Sum assertion fails for an already published optimum

This is a source/status correction, not a new-paper or priority claim. The odd-order part of Ahmed-Snevily's Odd Sum conjecture (Conjecture 2.4, p. 4) is contradicted by

pi = (8,14,3,15,6,10,2,17,7,12,1,16,5,11,4,13,9).

This permutation is already in Botler-Netto (2021), Table 1, with score 798783 in Table 2 (published paper). Netto's 2022 dissertation, Table 2.1, printed p. 25 (PDF p. 35) explicitly states that the unrestricted model proves the listed n = 14,15,16,17 examples optimal (institutional PDF). Thus pi belongs to RC(17), but, using one-based positions, pi_4 + pi_14 = 15 + 11 = 26 is even. Likewise pi_6 + pi_12 = 10 + 16 = 26.

The contradiction is an immediate consequence of that published optimum; I am not attributing an explicit Odd Sum discussion to those authors. A separate exact subset-DP check also gives max t = 798783, without assuming alternation or opposite-position symmetry. The statistic here counts maximal monotone contiguous runs, as in the source definitions.

Suggested annotation: mark the odd-order Odd Sum assertion false, while retaining the distinction from the enumeration conjecture and the even-order opposite-sum assertion, neither of which this notice resolves. In particular, the entire bundled record should not be labelled solved on this basis. The dated literature note saying that no counterexample to either displayed conjecture was verified misses this consequence of the existing table. AI-assisted, self-audited source check; no new discovery, independent review, or maintainer-applied change is claimed.

AIM-ALGEBRAIC_GEOMETRY-0181: known real solution counts, with exact reproducible examples

For AIM Algebraic Vision, Reconstruction, Problem 1.05, the answer on the stated ten-distinct-complex-solutions locus is 0, 2, 4, 6, 8 or 10 real projective essential matrices. In particular, all ten can be real. This is a source/status clarification, not a new-paper or priority claim.

The endpoints are already discussed in Agarwal, Lee, Sturmfels and Thomas (2015), Section 3.2, pp. 11-12: the ten-real case is attributed there to Maybank's Theorem 5.14, while Example 3.1 gives explicit integer correspondences with zero real solutions. Breiding, Fairchild, Santarsiero and Shehu, Figure 1, p. 4, numerically sample the intermediate counts as well. Their figure includes 0,2,4,6,8, not a sampled ten-real case.

For reproducibility, here are small-integer input pairs I checked using exact rational elimination and a separate standard-library Fraction verifier. In each line the five pairs in U correspond, in order, to the five pairs in V; append coordinate 1 to every pair and impose u_i^T E v_i = 0.

real  U                                      V
0     (-4,-1),(-1,2),(0,-2),(5,-4),(2,3)       (-4,-2),(4,-5),(-1,-4),(-2,-2),(-2,3)
2     (-2,-2),(0,5),(-4,-5),(-3,3),(-2,5)      (4,1),(-3,-3),(-5,5),(-1,4),(0,-2)
4     (5,0),(-5,-4),(4,-1),(4,-2),(2,0)        (-4,1),(0,5),(3,-1),(2,-3),(-2,-2)
6     (-2,-3),(1,3),(0,-4),(-5,0),(-3,-1)      (-3,5),(1,5),(-1,-2),(-3,3),(4,-1)
8     (0,-2),(0,1),(4,5),(4,-4),(0,-1)        (5,3),(-2,-1),(-5,-4),(-5,1),(5,2)
10    (0,-3),(0,-5),(-4,5),(1,1),(-4,0)       (-3,2),(-5,5),(-3,4),(4,4),(4,-5)

For each input, the five epipolar constraints have rank five. In a rational kernel chart E=xB0+yB1+zB2+B3, the ten essential cubics have an invertible coefficient block on all degree-three monomials. This excludes chart-infinity solutions and bounds the quotient dimension by ten. A squarefree degree-ten polynomial p(z) and rational coordinate polynomials x(z), y(z) satisfy all ten original cubics modulo p. Thus the roots give all ten simple complex solutions. Exact Sturm variation pairs at (-infinity,+infinity) are respectively (5,5), (6,4), (7,3), (8,2), (9,1), (10,0). Conjugation gives the even-count restriction. This uses classical elimination/Sturm methods, not a new algorithm.

Please distinguish this algebraic count from positive-depth camera-pose counts and from nontransverse degeneracies, which are not covered by the ten-simple-solutions premise. A geometric classification of all zero-real input configurations is a different question. AI-assisted and self-audited reconstruction; no independent peer review or maintainer-applied status change is claimed.

New scoped preprint for AIM-COMBINATORICS-0131 (not a complete resolution of the open-ended source item)

Alper Ferudun, "Critical Groups Detect Homology Actions of Simple Graphs" (4 pages): https://doi.org/10.5281/zenodo.23471711
PDF, source and exact-check scripts: https://eulersolve.org/papers/aim-combinatorics-0131/

Theorem: for every finite connected simple graph, the natural automorphism actions on the finite critical group and on real first homology have exactly the same kernel. This includes graphs with bridges and trees. Combining the equality with the known homology theorem gives a faithful critical-group action whenever the minimum degree is at least two and the graph is not a cycle. For a cycle, the kernel is its rotation subgroup. Parallel edges invalidate the equality: a transposition of edges in a two-vertex banana graph fixes all vertex divisors but moves an integral flow.

The proof uses the equivariant cut-flow presentation and the fact that a nonzero integral flow in a simple graph has squared norm at least three. Edge reversal requires a separate coordinate argument; a blanket norm-at-most-two claim for all edge differences would be incorrect. Two exact implementations agree on the 577 bridgeless atlas graphs through seven vertices; the independent implementation checks the kernel equality for all 995 connected graphs and 13,286 automorphisms. These finite checks supplement, not replace, the general proof.

Attribution and scope: the cut-flow description and homological faithfulness are established results, credited in the manuscript. In particular, Estélyi et al., arXiv:2102.04239v3, Proposition 5 supplies the minimum-degree homology result (https://arxiv.org/abs/2102.04239v3). The current dataset's weaker three-edge-connected faithfulness reduction is not claimed as a new discovery. The comparison also includes the 2025 paper https://doi.org/10.1016/j.disc.2024.114259; any consequence for its embedding question is explicitly restricted to simple graphs.

Levine's original AIM item 41 (https://aimath.org/pastworkshops/chipfiringproblems.pdf) is a research suggestion about the critical group with a harmonic action, not a fixed yes/no conjecture. This paper proves one precise general theorem within that suggestion. Our source-record classification therefore remains PARTIAL_RESULT, with zero whole-record closure increment. The preprint is AI-assisted, self-audited and unrefereed. No independent review, formal certification, absolute priority, dataset change or maintainer acceptance is claimed.

AMR-042-0006: the scalar logarithm prescription cannot be a topological conjugacy invariant

This is a source/status clarification using known conjugacy results, not a new-paper or priority claim. Ward's Problem F, p. 3 asks for a meaningful p-adic entropy with h_p(L_lambda) = log_p(lambda), suggesting topological conjugacy invariance as an example requirement. That particular conjunction is impossible.

Here is a direct all-prime counterexample. Put u = 1+p^2 and define on Q_p

H(0) = 0,
H(x) = u^(v_p(x)) x  (x != 0).

H preserves each valuation shell and has inverse x -> u^(-v_p(x)) x. Within one shell it multiplies differences by a unit; between different shells the ultrametric distance depends only on the two valuations. Thus H is a global surjective isometry. Moreover, H(px) = pu H(x), so multiplication by p and by pu are conjugate. Their prescribed values are respectively 0 and log_p(1+p^2) != 0: the first term of the logarithm series has valuation 2, strictly smaller than every later term.

This conjugacy obstruction is already implied by Jenkins-Spallone (2010), Section 7, Proposition 7.1, p. 26. The application to Ward's prescription is an immediate deduction, not an explicit entropy theorem attributed to those authors.

The compact Haar-probability setting does not rescue it. On Z_2 write each nonzero x uniquely as 2^k r t, with r in {1,3} and t in 1+4Z_2. The map G(0)=0, G(2^k r t)=2^k r t^3 is a surjective isometry conjugating multiplication by 5 to multiplication by 125. Both multipliers and G preserve Haar probability, but their logarithms differ by 2 log_2(5) != 0.

Suggested annotation: the topological-conjugacy-invariant scalar formula is impossible, rather than an open construction target. The broader request for other meaningful p-adic entropy notions is not closed by this observation. In particular, Deninger's periodic-point entropy and Mahler-measure theorem use different requirements and are not contradicted. AI-assisted, self-audited reconstruction; no independent review or maintainer-applied change is claimed.

OWR-13490-013: a literal counterexample and the neighbourly characterization

Alper Ferudun, Component Symmetry and Tightness in Dimension Two (4-page preprint):
https://doi.org/10.5281/zenodo.23477149
PDF, source and reproducibility files: https://eulersolve.org/papers/owr-13490-013/

Literal answer: no. The disk with facets {a,b,z1} and {a,b,z2} has symmetric link-component counts in every induced subcomplex but is not tight: the two nonadjacent vertices z1,z2 induce a disconnected pair whose H0 map into the disk is not injective. More generally, the same construction works for any number of at least two triangles sharing one edge.

Repaired theorem: for every finite nonempty simplicial complex X of dimension at most two, X is tight over a characteristic-two field if and only if its one-skeleton is complete and c_Y(u,v)=c_Y(v,u) for every induced subcomplex Y and distinct vertices u,v. In particular, tightness over any field implies characteristic-two tightness in this dimension.

The proof adapts the cycle-link path argument in Bagchi, Datta and Spreer, Tight triangulations of closed 3-manifolds, Lemmas 3.2–3.4, https://doi.org/10.1016/j.ejc.2015.12.006 . An induced cycle in a vertex link produces an induced closed surface; its mod-two fundamental chain fills that cycle in the vertex deletion. Mayer–Vietoris then proves tightness. The prior lemmas and method are explicitly credited, not presented as new.

Source check: the conjecture on printed p. 299 of the 2015 Oberwolfach report itself omits neighbourliness. The current dataset statement does contain the mathematical assertion, despite its dated literature assessment saying the assertion was omitted. Suggested annotation: distinguish the negative answer to the literal wording from the positive theorem with neighbourliness restored.

Two exact implementations agree through five vertices; the C++ enumeration covers all 1,048,576 labelled triangle subsets on six vertices with complete one-skeleton. The independent Python verifier performs 23,616 checks, including explicit surface fillings and book counterexamples. These checks supplement, not replace, the proof. This is one manuscript and one literal source-record negative answer, not a solution of higher-dimensional field independence. AI-assisted, self-audited and unrefereed; no independent review, formal certification, absolute priority, dataset change or maintainer acceptance is claimed.