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Updated Oct 8

Problem 174

claims/: The 6 claim pages of Problem 174, one per claimant's result; the problem's standing derives from them.

Statement. A finite set $A\subset \mathbb{R}^n$ is called Ramsey if, for any $k\geq 1$ , there exists some $d=d(A,k)$ such that in any $k$ -colouring of $\mathbb{R}^d$ there exists a monochromatic copy of $A$ . Characterise the Ramsey sets in $\mathbb{R}^n$ .

Notes. The site asks to "characterise the Ramsey sets". The classification accepted here is a criterion: a finite set is Ramsey exactly when an explicit linear system over the tensor square of its coordinate field has a solution. That is a complete if-and-only-if description of the Ramsey sets, so the problem is recorded as solved. It is not a geometric description of the kind Graham or Leader, Russell and Walters conjectured (both of which the paper refutes, the first in Lean built here, the second as the release's own claim), and no procedure for deciding the criterion from numerical data is proved. The site's page (last edited 16 October 2025) keeps the label OPEN and its curator has not commented on the classification or on Pálvölgyi's heptagon; the standing here would follow the curator's ruling if the curator reads 'characterise' more narrowly.

Formulation. A copy is a congruent copy at the original scale, as Erdős, Graham, Montgomery, Rothschild, Spencer and Straus define it and as the claim pages read it; allowing a change of scale would be a different question.

Status. OPEN, in the site's label (page last edited 16 October 2025; proof-claims thread with no claim as of 6 October 2026); the site's remarks record neither result of September 2026, though Pálvölgyi announced the heptagon in the problem's discussion on 23 September 2026. The derived standing departs from the label because the classification of September 2026 postdates the site's last edit and is accepted here on its Lean proof. The standing in the frontmatter derives from the claim pages. OpenAI's classification of 23 September 2026, recorded on its claim page, answers the question: a finite set is Ramsey exactly when an explicit system of linear equations over the tensor square of the field generated by its coordinates has a solution; the proof is kernel-checked in Lean, built and axiom-audited here, which is the acceptance evidence its claim page lists, and no referee or outside reviewer has examined it. No decision procedure for that criterion is proved. Pálvölgyi's seven-point circle configuration of 20 September 2026, recorded on its claim page, is a claimed, unreviewed partial result against Graham's conjecture that every spherical set is Ramsey; the accepted classification refutes that conjecture independently through its twelve-point set, built in Lean here. The classification paper also claims to refute the Leader–Russell–Walters conjecture, through its five-point Corollary 7.4 together with Corollary 2 of Leader, Russell and Walters's 2011 paper; the corpus has not built or reviewed that step, so that refutation stands as the release's claim. Four refereed results that decide the classification for classes of sets are recorded as accepted partial claims, each on the refereed publication alone: sphericity as a necessary condition and boxes as Ramsey, on EGMRSS 1973; nondegenerate simplices, on Frankl and Rödl 1990; subsoluble sets, regular polygons and the regular polyhedra, on Kříž 1991; and cyclic trapezoids, on Kříž 1992. The earlier results below give necessary conditions, positive classes and closure operations, and the August 2026 pyramid papers describe the classification as unresolved at their date.

Source. T. F. Bloom, Erdős Problem #174, accessed 5 September 2026; the page was last edited on 16 October 2025, so its progress list omits several later papers.

Formalization. The classification is kernel-checked in the Lean development of OpenAI's release, pinned and described on the claim page; the corpus built the four declarations it rests on and checked their axioms. The "Current assessment" section below records the earlier statement search and the public statement files.

Current assessment

Two results of September 2026 come first; the earlier literature follows. OpenAI's preprint A classification of finite Euclidean Ramsey configurations (23 September 2026) proves that a finite set of at least two points, taken as a congruent representative with full affine span in its affine dimension, is Ramsey if and only if a stated $(d+1)\times(d+1)$ matrix equation has a solution over $F\otimes_{\mathbb Q}F$ , where $F$ is the field generated by the coordinates; multiplying the tensors out recovers a sphere equation, so the criterion sharpens sphericity. The classification, and the twelve-point spherical set that fails the test, are kernel-checked in Lean and built here, and its claim page states the theorem, the declarations and what the corpus checked; the paper's further consequences, that every subtransitive set and every set of at most five concyclic points passes the test and that nine generic concyclic points fail it, are stated in the manuscript and come with Lean declarations in the release, pinned by its comparator challenges, which the corpus has not built or audited, so they stand as the release's claims. The test is exact arithmetic in $F$ and no decision procedure for it is proved. Three days earlier, Pálvölgyi's cyclic heptagon was claimed to be the first spherical set that is not Ramsey, by a derivation-weighted version of the EGMRSS argument; that proof is unreviewed, and its claim page records it as claimed, with the author's attribution of the proof to ChatGPT. Neither result is in the site's remarks or in a journal; Pálvölgyi announced the heptagon in the site's discussion of the problem on 23 September 2026.

Search scope. The search checked primary arXiv records, source manuscripts, publisher records and research-announcement leads beyond erdosproblems.com. The Moore, Mirabi and Ivan–Leader–Walters prism manuscripts have complete author-recorded reconstructions here, which are not journal acceptance. No acceptance or published version for these manuscripts or Pálvölgyi's near-circumsphere paper (arXiv:2608.10865) was located. Its v2, p. 24, cites both pyramid proofs, providing dated primary uptake. Neither uptake nor an announcement substitutes for proof review.

The site's February 2026 discussion points to Behague's Nearly all known Euclidean Ramsey sets are subsoluble. The selected version is v3, 4 December 2025; the revision history says v2 replaced the simplex argument with a citation to Karamanlis's earlier proof. An indexed record of the journal's accepted-papers list listed the paper as accepted for publication; the author's page listed it under Submitted. No published journal article was located. Acceptance evidence, publication and independent proof review are separate. Behague's selected v3 has a complete author-recorded source chain at its stated external inputs, including the constructions below. The version and correction record preserves the deleted v1 simplex argument without claiming a complete proof of it; the replacement Karamanlis construction is compiled separately. The paper's five source-dated questions record the remaining classification implications without claiming a new current-status review of each question.

The compiled [[../library/discrete_geometry/ivan_2026_block_sizes_block_sets_conjecture/_index|Ivan–Leader–Walters block-size paper]] uses the published 2026 article alongside its 2024 arXiv version. Its [[../library/discrete_geometry/ivan_2026_block_sizes_block_sets_conjecture/theorem_2|obstruction theorem]] shows that no single positive block-size bound works across all templates, even over three letters. Its [[../library/discrete_geometry/ivan_2026_block_sizes_block_sets_conjecture/theorem_3|degree-two theorem]] gives an optimal bound for template $123$ , for every number of colors. These results do not prove the full block-sets conjecture. The source's auxiliary assertion about powers of the regular hexagon contracted by $1/\sqrt2$ is refuted by the compiled [[../library/discrete_geometry/ivan_2026_block_sizes_block_sets_conjecture/geometric_power_scope|fixed-scale counterexample]]. That assertion is separate from the ordinary Ramsey property of regular hexagons and is not an input to either main block-size theorem.

The problem site reports no formalized statement. The formal-conjectures tree at its revision of 4 September 2026 has no 174.lean at the conventional path. That revision predates the release of 23 September 2026; the Lean development of that release, built here, is recorded on the accepted claim page.

The fixed-dimensional, two-color questions in #188 and #214 use related constructions, but their bounds do not answer this all-color classification.

Necessary condition and the classification questions

Every Ramsey set lies on a sphere. The complete proof is Erdős–Graham–Montgomery–Rothschild–Spencer–Straus, Theorem 13, with its affine-relation criterion and field-coloring prerequisites; the accepted partial claim EGMRSS 1973 records it with the product theorem below. Its avoiding color count is independent of the ambient dimension. For example, three distinct collinear points are nonspherical and hence are not Ramsey in the all-color sense, even though two colors in $\mathbb R^3$ force every prescribed three-point configuration. The latter is the different, fixed-color Theorem 8.

Graham's conjecture is that being spherical is sufficient. A rival conjecture of Leader, Russell and Walters says that the Ramsey sets are exactly the subtransitive sets: those congruent to subsets of finite Euclidean configurations whose isometry groups act transitively. Their paper proves that some spherical sets are not subtransitive, so the two proposed characterizations differ. These conjectures and the separation result are attributed here to their 2010 manuscript, published in JCTA 119 (2012), 382–396; its separation proof is compiled. The distinct 2011 cyclic-quadrilateral construction gives explicit four-point examples of the same separation. Graham's conjecture is refuted by the classification paper's twelve-point set, built in Lean and accepted here, which Pálvölgyi's claimed heptagon of three days earlier had already contradicted. The same paper claims to refute the Leader–Russell–Walters conjecture through its Corollary 7.4, that every nonempty set of at most five concyclic points is Ramsey, which would make the non-subtransitive cyclic kites of Corollary 2 of their 2011 paper Ramsey; that corollary is not among the declarations the corpus built, and the step is unreviewed, so that refutation stands as the release's claim. The classification replaces both proposed geometric descriptions with an algebraic test; see the claim pages linked above.

A subsoluble set embeds in a finite configuration admitting a transitive soluble, or solvable, isometry group. Kříž's theorem proves that every such set is Ramsey. Subsolubility supplies a sufficient condition; it is not a known complete characterization. See the soluble-group theorem and subset closure.

Positive classes and reusable proof methods

Products and bricks. Finite orthogonal products of Ramsey sets are Ramsey. Starting with two-point sets gives every finite rectangular box and its subsets. The product proof uses the cardinality of a finite witness in its color-pattern count (claim page: EGMRSS 1973); the brick consequences and compactness step are recorded separately.
All finite nondegenerate simplices. Frankl and Rödl's 1990 theorem proves the stronger exponential finite-witness density property (claim page: Frankl and Rödl 1990). The ordinary Ramsey consequence follows by a largest-color-class argument. The complete same-paper chain is compiled, with its precise external joint-partition input (Theorem 1.16), whose original 1987 proof chain is compiled separately. Karamanlis's distinct polygonal-torus construction embeds every simplex in a finite product of regular polygon vertex sets. Independent rotations give a transitive abelian enclosure, so the Kříž consequence recovers ordinary Ramseyness. This route supplies neither the exponential density bound nor a prescribed containing radius. Its complete chain includes the explicitly repaired approximation lemma.
Soluble symmetry and two-orbit extensions. Kříž's transitive soluble-group theorem and two-orbit theorem give further positive classes. In particular, regular polygons and the five regular convex polyhedra are Ramsey (claim page: Kříž 1991). The orbit-gluing argument and its source corrections have their own proof pages. Behague's stronger subsoluble-enclosure theorem covers every finite convex regular polytope except the 120-cell and 600-cell. Its dodecahedron construction uses the precise subgroup-index hypothesis of the two-orbit enclosure lemma. Those two exceptions are exclusions from this theorem, not assertions that they are non-Ramsey.
Four coordinate-permutation families. Behague's Theorem 1.6 constructs subsoluble enclosures for the permutations of $(\alpha^i,\beta^j)$ , $(\alpha^i,\beta^j,\gamma)$ , $(\alpha^i,\beta^j,\gamma,\gamma)$ and $(\alpha^i,\beta^j,\gamma,\delta)$ , where the exponents denote coordinate repetition, $i,j$ are nonnegative integers and the real parameters need not be distinct. Its signed-coordinate wreath action proves these four families, not the full block-sets conjecture.
Cyclic trapezoids. Kříž's 1992 theorem applies to a quadrilateral inscribed in a circle with two parallel sides. That cyclic hypothesis is part of the source's definition of “trapezoid”; the title is not a claim about arbitrary nonspherical trapezoids. The published abstract gives this exact scope (claim page: Kříž 1992). Its four-page proof is not reconstructed in the library, unlike the compiled 1991 group paper. Behague's complete alternative enclosure proof lowers the trapezium's altitude into a finite dihedral orbit and restores the original distances using two product layers. It proves subsolubility for nondegenerate convex reflection-symmetric isosceles trapezia, including rectangles. This supplies another proof for the cyclic class; it does not reconstruct the unavailable 1992 argument.
Generalized prisms. Ivan, Leader and Walters's 2026 theorem constructs a subtransitive prism from two finite layers on which one common finite isometry group acts transitively, with any nonzero perpendicular separation $\lambda\ne0$ . If that group is soluble, the prism is subsoluble and therefore Ramsey. Merely having isomorphic symmetry groups does not supply the common-action hypothesis.
An arbitrary off-affine-hull extension. If $B$ is finite and Ramsey and $z\notin\operatorname{aff}(B)$ , then $B\cup\{z\}$ is Ramsey. Moore's August 2026 proof uses a product with an auxiliary simplex; Mirabi's distinct proof uses cyclic constructions and Kříž's equivalence-relation machinery. Moore's complete source chain is compiled; Mirabi's proof is sketched on its result pages. No symmetry or convexity hypothesis on the Ramsey base remains in the final conclusion.

Prescribed spheres and their limits

For a nonempty spherical set, its intrinsic circumradius $\rho$ is the radius about the unique equidistant center in its affine hull. This need not be the radius of its smallest enclosing ball. Frankl and Rödl's 2004 simplex theorem provides exponentially dense finite witnesses on every sphere with squared radius $\rho^2+\alpha$ , for each $\alpha>0$ , in every sufficiently large ambient dimension. Its color consequence therefore forces the simplex on every strictly larger prescribed sphere. The proof record includes the corrected near-regular radius budget and exact dimension-padding steps. Its deep external inputs remain explicit.

Pálvölgyi's 3 September 2026 revision, Definition 1 and Theorem 2 (card, for the 11 August 2026 v1) claims the near-circumsphere conclusion for every finite configuration $P$ admitting a transitive solvable isometry group: for every integer $r\ge1$ and $\varepsilon>0$ , some sphere $S^n_{\rho(P)+\varepsilon}\subseteq\mathbb R^{n+1}$ forces a monochromatic congruent $P$ under every $r$ -coloring. Its full main proof is not reconstructed in the library. The statement does not remove solvability, automatically apply to subsoluble subsets at their own intrinsic radius, or include the zero-slack endpoint. The added appendix and author qualifications are unverified leads: the author expressly says that the author has not independently verified every appendix claim. Those claims are not used to settle #174.

Radius control must be checked before restricting a containing configuration to a subset. In particular, the compiled 1990 Corollary 6.5 record keeps its general smaller-intrinsic-radius subset clause unfilled. The 2004 simplex proof does not silently close that separate gap.

Canonical Ramsey comparison

The distinct canonical Ramsey condition asks for one finite host such that every coloring, with any palette, contains a congruent target that is monochromatic or rainbow (all its points have different colors). Shaw's Theorem 2 (arXiv v1, 19 August 2026) proves that for every prime $p$ , every integer $k\ge1$ and every regular $p$ -gon $C$ , there is an integer $n\ge1$ with

p^{-p/2}C^n\longrightarrow_{\mathrm{MR}}C^k.

Here $C^k$ is an orthogonal product, $p=2$ means two distinct points, and the scale belongs to the host; the target is a congruent, unscaled copy of $C^k$ . The host is chosen before the coloring and its palette.

For composite $q\ge6$ , the compiled polygon-power obstruction gives, on every $C_q^n$ with $n\ge1$ , a coloring excluding every scaled monochromatic or rainbow copy of the regular $q$ -gon $C_q$ . Thus no positive dilation of such a power is a canonical witness for $C_q$ . This concerns the specified host family; it neither rules out a different canonical witness nor gives a negative ordinary Ramsey result.

Linked library material

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