Openai 2026 classification finite euclidean ramsey configurations

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corollary_7_2: The manuscript's claimed proof of the sufficiency half of the Leader--Russell--Walters subtransitive conjecture, by averaging the squared affine coordinate rows of the transitive set over its isometry group to build a tensor certificate for Theorem 1.1; claims checked, not independently reviewed.

corollary_7_4: The manuscript's small-concyclic-set theorem, derived from Proposition 7.3 (linear independence of the quadratic evaluation rows of a spherical set implies the tensor criterion) by interpolating with products of two line equations; in particular every cyclic quadrilateral is claimed Ramsey; claims checked, not independently reviewed.

corollary_7_5: The manuscript's claimed counterexample to the necessity direction of the Leader--Russell--Walters subtransitive characterization and to their kite conjecture: Corollary 7.4 makes the kite Ramsey, and their 2011 Corollary 2 is cited for non-subtransitivity; claims checked, not independently reviewed.

theorem_1_1: The manuscript's classification: a finite set of at least two points spanning R^d is Ramsey at fixed scale if and only if some matrix over the tensor square of its coordinate field has zero evaluations at every point and multiplied spatial block the identity; claimed resolution of Problem 174, read at claims-checked depth, not independently reviewed.

OpenAI, A classification of finite Euclidean Ramsey configurations, OpenAI Math Release preprint, September 23, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_classification_finite_euclidean_ramsey_configurations.pdf, and the release's TeX bundle sits beside paper.pdf in that folder.

@misc{OAI:A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026,
  author = {{OpenAI}},
  title = {{A classification of finite Euclidean Ramsey configurations}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf}{OAI:A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026}},
  year = {2026}
}

Attestation, recorded as the source's own statements and not as this corpus's review: the release's root README says its manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all have Lean formalizations and that "Some of the unformalized results could have issues". The manuscript's own README adds only the title, the author line "OpenAI", the date and the citation block; it carries no statement on human assistance or review. The text of the manuscript names no author, affiliation beyond the title page, arXiv identifier or journal. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

Formalization, as the release lists it: lean/formalization.yaml names this manuscript, and the release's family page says its formalization gives the tensor-field classification for configurations whose affine span is the whole space, covers the singleton and affine-span reductions, proves that every Ramsey configuration is cospherical, that every nonempty subset of a finite transitive configuration and every nonempty set of at most five circle points is Ramsey, a sufficient condition from linear independence of quadratic evaluation rows, and two spherical non-Ramsey examples (the twelve-point set with a fifty-color obstruction in every positive dimension and the nine-point circle configuration from algebraically independent parameters). The comparator statement files it names are EuclideanRamsey.lean (classification), GrahamSpherical.lean (twelve points), EuclideanRamseyCircle.lean (five circle points), EuclideanRamseyNine.lean (nine points), EuclideanRamseyQuadratic.lean (quadratic independence), EuclideanRamseySpherical.lean (cosphericity) and EuclideanRamseyTransitive.lean (subsets of transitive sets), all under lean/ComparatorChallenges/; the comparator table names no file for the reductions. The corpus's verification built the declarations OAI.EuclideanRamsey.classification_nonempty, OAI.EuclideanRamsey.classification and OAI.EuclideanRamsey.quadratic_empty_ramsey (the classification, for every nonempty finite set and for the empty set) and OAI.GrahamSpherical.full_main (the GrahamSpherical.lean statement, the twelve-point example) and checked their axioms (propext, Classical.choice and Quot.sound only); the record of what they settle is kept on the claim page of Problem 174. The other comparator statements are read statically from the release's catalogue; not built, replayed or audited for fidelity in this repository.

Companions: the release lists no other manuscript in this manuscript's family.

Read status: claims checked for Theorem 1.1, Corollary 7.2, Corollary 7.4 and Corollary 7.5, and for the statements of Propositions 2.1, 2.4, 7.3, 7.6 and 7.7 and Corollary 7.1, read clause by clause in the TeX source (sections/01-introduction.tex lines 74--88; sections/02-tensors.tex lines 33--36 and 144--149; sections/07-consequences.tex lines 8--10, 29--31, 62--71, 106--109, 124--131, 153--161 and 210--224) on 2026-10-07; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

• Section 1, The classification problem (sections/01-introduction.tex; PDF pp. 1--4). Defines a finite nonempty set $A$ to be Ramsey when for every $r\ge2$ some $\mathbb R^D$ has a monochromatic congruent copy of $A$ under every map $c:\mathbb R^D\to\{1,\ldots,r\}$, congruence preserving every distance at the original scale and the coloring arbitrary (no measurability). Recounts the sphericity necessity of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus (Theorem 13 of their 1973 paper), Graham's sphericity conjecture, Frankl--Rödl (triangles, simplices), Kříž (soluble transitive groups, cyclic trapezoids), Cantwell (regular polytopes), the Leader--Russell--Walters subtransitive conjecture (Conjecture A of their 2012 paper) and the block-sets program, Karamanlis, Behague, Ivan--Leader--Walters prisms, the Moore and Mirabi one-point extension theorems, and Pálvölgyi's September 2026 seven-point circle configuration claimed to disprove the spherical conjecture (with the manuscript's footnote that Pálvölgyi attributes that work to ChatGPT and labels its appendix claims unchecked). Sets up, for a set of $s\ge2$ points affinely spanning $\mathbb R^d$, the columns $p_i=(1,a_i)^{\mathsf T}$, the coordinate field $F=\mathbb Q((a_i)_\alpha)$, the ring $B=F\otimes_{\mathbb Q}F$ and the multiplication map $m_F:B\to F$, and states Theorem 1.1 (Classification): $A$ is Ramsey if and only if some $P\in\operatorname{Mat}_{d+1}(B)$ has $(p_i\otimes1)^{\mathsf T}P(1\otimes p_i)=0$ for every $i$ and $m_F(P_{\alpha\beta})=\delta_{\alpha\beta}$ on the spatial block. States that the condition is representative-independent and uses exact coordinate-field relations, not numerical approximations. Previews the consequences and the three-stage sufficiency proof. Ends "All arguments take place in ZFC."
• Section 2, Rational tensors and the coloring obstruction (sections/02-tensors.tex; pp. 4--7). Proposition 2.1 restates the matrix condition as the existence of a flip-invariant tensor $T\in W\otimes_{\mathbb Q}W$, $W=\mathbb R\times\mathbb R^d$ viewed over $\mathbb Q$, with $(e_i\otimes e_i)(T)=0$ for the evaluations $e_i(v,u)=v+a_i\cdot u$ and gradient Gram matrix $G(T)=I_d$; the proof symmetrizes and descends from $\mathbb R\otimes_{\mathbb Q}\mathbb R$ to $B$ by a finite linear system over $F$. Remark 2.2 notes $B$ is Noetherian, so the kernel of the evaluation system is finitely generated; Remark 2.3 argues congruence invariance. Proposition 2.4 (necessity): if no such tensor exists, a $\mathbb Q$-linear functional separating $(0,\ldots,0,I_d)$ from the image yields functions $h_i$ with $\sum_i h_i(v+a_i\cdot u)+L(uu^{\mathsf T})=0$, $L(I_d)=1$, and a coloring of every $\mathbb R^D$ by the residues of $H_i(z)=\sum_j h_i(z_j)$ modulo $2$ in $K=2s+1$ intervals, with $r_0=K^s$ colors independent of $D$, avoids every congruent copy. The manuscript places this in the invariant-and-coloring tradition of the 1973 paper's Theorem 13 and Lemma 15 and Rado's 1945 note.
• Section 3, A finite lattice identity (sections/03-stencil.tex; pp. 7--9). Lemma 3.1: given symmetric $C\in\operatorname{Mat}_k(\mathbb Q)$ whose tensor vanishes in each $(\mathbb Q^k/V_i)^{\otimes2}$, there is a finitely supported $\nu:\mathbb Z^k\to\mathbb Q$ with zero sum on every coset of every $\Lambda_i$, zero mass and first moment, and second moment $C$. The proof works in the Laurent polynomial group algebra, passes to its $\mathfrak m$-adic completion in formal logarithmic coordinates, and uses flatness of the completion of a Noetherian ring (Stacks Project Tag 00MB, cited) to intersect the ideals before truncating back to a finite Laurent polynomial.
• Section 4, Exact coordinate permutations at two scales (sections/04-pairs.tex; pp. 9--13). Lemma 4.1 (Two-scale configurations): from a tensor certificate, for every $\eta>0$ there are vectors $f_i,g_i\in\mathbb R^\ell$ with $\|f_i-f_j\|^2=\|a_i-a_j\|^2$, $\|g_i-g_j\|^2=q\|a_i-a_j\|^2$, $q=\eta/(1+\eta)$, each $g_i$ a coordinate permutation of $f_i$. Inputs: Lemma 4.2 (a compactly supported smooth $b\ge0$ whose $(D_Cb)_-$-weighted gradient second moment is small, built by averaging Gaussian densities with covariances $S+tC$ and cutting off), Lemma 4.3 (discretization by convolution of $\nu$ with a scaled copy of $b$, then rational approximation inside an open set where the strict inequalities persist). The proof of Lemma 4.1 splits the signed weights into positive and negative lists of affine rows, appends common rows to fix the Gram matrices at $(1+\eta)I_d$ and $\eta I_d$, and rescales.
• Section 5, From paths to monochromatic copies (sections/05-paths.tex; pp. 13--17). In the free group $K$ on symbols $\xi_z$, $z$ in the finite-support sequence space $\mathcal E$, a path is a product of diagonal factors (weight $0$) and factors $X_b$ recording an ordered copy of $A$ scaled by $\sqrt u$ (weight $u$). Proposition 5.1 (path criterion): if for every $n$ two paths share an endpoint with weight ratio below $1/n$, then $A$ is Ramsey. Lemma 5.2 is the compactness reduction from $\mathcal E$ to a finite $\mathbb R^D$ (compare Proposition 4 of the 1973 paper, reproved for arbitrary colorings); Lemma 5.3 synchronizes one endpoint with paths of weights $1,1/2,\ldots,1/n$ by concatenating rescaled copies; Lemmas 5.4 and 5.5 build the Stone--Čech semigroup $\beta\mathcal S$ of finite strings, an idempotent $p$ whose corner $p(\beta\mathcal S)p$ is a group (Hindman--Strauss and Ellis cited for the convention; proved inline), and the simultaneous realization of ultrafilter products by strings with common factors realized identically. The proof of Proposition 5.1 colors words of a Hales--Jewett length $n$ through the homomorphism $K\to p(\beta\mathcal S)p$, lets a monochromatic line choose its number $t$ of variable positions, and expands each variable position with the weight-$1/t$ path so that the squared-distance factor is $t\cdot(1/t)=1$. Figure 1 (p. 17) records the order of these choices.
• Section 6, From coordinate permutations to equal path endpoints (sections/06-groups.tex; pp. 17--21). Lemma 6.1: the projection of the unit-copy monoid $M$ to any two coordinates is a group containing $N\times N$, $N$ the augmentation kernel of $K$. Lemma 6.2: the single-coordinate correction sets $L_i$ are normal subgroups with a subadditive, conjugation-invariant cost, and $N^{(s-2)}\subset L_i$ (iterated commutator subgroup). Lemma 6.3: averaging over a finite coordinate permutation group drives within-orbit discrepancies into $L=N^{(s-2)}$ with cost at most $\delta$ times the degree. Proposition 6.4 turns the two images of Lemma 4.1 into equal-endpoint paths with weight ratio at most $q+s\delta$. The completion of sufficiency (p. 21) chains Proposition 2.1, Lemma 4.1 with $q<1/(2n)$, Proposition 6.4 with $\delta=1/(2sn)$ and Proposition 5.1; the manuscript notes that every choice is made after fixing $n$ and no uniform bound is needed.
• Section 7, Geometric consequences and examples (sections/07-consequences.tex; pp. 21--25). Corollary 7.1 rederives sphericity of Ramsey sets by applying $m_F$ to the evaluation equations. Corollary 7.2: every nonempty subset of a finite transitive Euclidean set is Ramsey, by averaging the squared affine coordinate rows over the finite isometry group. Proposition 7.3: a spherical set whose $s$ rows $(p_{i\alpha}p_{i\beta})$ are linearly independent over $F$ is Ramsey (an adjugate lift of an $F$-solution to the tensor ring). Corollary 7.4: every nonempty concyclic set of at most five distinct points is Ramsey, via products of two line equations (the manuscript notes the same interpolation in Pálvölgyi's Theorem A.2). Corollary 7.5: for transcendental $a\in(-1,1)$ the kite $K_a$ is Ramsey and not subtransitive, the second part cited to Leader--Russell--Walters (2011), Corollary 2; the manuscript concludes that the necessity direction of Conjecture A and the kite non-Ramsey conjecture (their Conjecture 3) fail. Proposition 7.6: nine circle points with algebraically independent parameters are spherical and not Ramsey (a $9\times9$ determinant specialized to a Kronecker square with determinant $64$; the manuscript says Pálvölgyi's generic seven-point claim, labeled unchecked by its author, already implies this). Proposition 7.7: the twelve-point set formed by the square $Q_0=\{(\pm1,0),(0,\pm1)\}$ together with its rotations by $\pm2\arctan t$, $t$ Liouville's constant, lies on the unit circle and is not Ramsey, by a derivation on $\mathbb Q(t)$ and weighted moment identities (transcendence of $t$ proved inline, Liouville 1851 cited).
• References (pp. 25--26): 23 printed entries, including Ellis 1958, Hales--Jewett 1963, Hindman--Strauss 1998, Shelah 1988, Rado 1945, the Stacks Project, and the 2025--2026 arXiv manuscripts of Behague, Ivan--Leader--Walters, Moore, Mirabi and Pálvölgyi (arXiv:2609.23327v1). The TeX bibliography file holds three further entries that the body never cites (Eberhard 2013, Pálvölgyi arXiv:2608.10865v2 and Shaw arXiv:2608.19183v1).

External inputs the proofs rest on, at statement level: the Hales--Jewett theorem (existence of the length only); flatness of the $\mathfrak m$-adic completion of a Noetherian ring (Stacks Project Tag 00MB); Leader, Russell and Walters (2011), Corollary 2, for the non-subtransitivity half of Corollary 7.5. The ultrafilter semigroup facts and Liouville's transcendence are proved inline with citations for the convention. The manuscript flags nothing as numerical, computer-assisted or conditional; it states that the criterion is exact in the coordinate field and not a numerical procedure, and that its arguments take place in ZFC.

Bears on

• Problem 174: claimed resolution. The problem asks for a characterization of the finite Ramsey sets at the original scale under every finite coloring; Theorem 1.1 states a necessary and sufficient condition for exactly that property (singletons trivially Ramsey, every other set reduced to its affine span), and Corollaries 7.2 and 7.4 and Propositions 7.6 and 7.7 claim to place the subtransitive sets, the small concyclic sets and two spherical examples on the two sides of it. Propositions 7.6 and 7.7 claim explicit spherical sets that are not Ramsey; if correct, they refute the sufficiency of sphericity, which the page records as Graham's conjecture and keeps as one of two rival characterizations (the manuscript also reports Pálvölgyi's seven-point configuration, arXiv:2609.23327v1, as already claiming this). The proofs were read for structure only. The corpus's verification built OAI.EuclideanRamsey.classification_nonempty, OAI.EuclideanRamsey.classification and OAI.EuclideanRamsey.quadratic_empty_ramsey, which state the classification, and OAI.GrahamSpherical.full_main, the twelve-point spherical example, and checked their axioms (propext, Classical.choice and Quot.sound only); the record of what they settle is kept on the claim page of Problem 174. The release's other comparator statements are not built or audited here.
• Leader--Russell--Walters Conjecture A: the manuscript claims both halves of that page's statement A are settled in opposite directions, the sufficiency half (subtransitive implies Ramsey) proved as Corollary 7.2 and the necessity half refuted by Corollary 7.5; unverified here, and that page's record of the conjecture as open stands until acceptance evidence is recorded.
• Leader--Russell--Walters Conjecture 3: claimed refutation. Corollary 7.5 asserts the kite of that conjecture is Ramsey for every transcendental $a\in(-1,1)$, using that paper's Corollary 2 only for non-subtransitivity; unverified here.