we-solve (beta)

Math-2276 · mathematics · Combinatorics

A four-point Hamming barrier in Euclidean Ramsey theory

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
A finite subset $X$ of Euclidean space is Ramsey if, for every number of colors $r$, some dimension $N$ forces a monochromatic congruent copy of $X$ in every coloring of $\mathbb R^N$. Is every finite set lying on a sphere Ramsey?

Formulation references: [S1], [S2].

Short English statement

A finite subset $X$ of Euclidean space is Ramsey if, for every number of colors $r$, some dimension $N$ forces a monochromatic congruent copy of $X$ in every coloring of $\mathbb R^N$. Is every finite set lying on a sphere Ramsey?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2276-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2276

Finite Euclidean Ramsey classification: The classification paper exhibits finite spherical configurations that are not Ramsey, directly refuting the universal spherical-implies-Ramsey claim.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

A four-point Hamming barrier in Euclidean Ramsey theory

OpenAI's claim

Finite Euclidean Ramsey classification: The announced characterization of finite Euclidean Ramsey sets is a tensor condition over the coordinate field, at the original congruence scale. Consequences include Ramsey behavior for subtransitive sets and small circular sets, as well as circular Ramsey examples that are not subtransitive.

Mathematical statement of the claimed result

Mathematical claim 1.
For distinct points $A=\{a_1,\ldots,a_s\}\subset\mathbb R^d$ with $s\ge2$ and affine span $\mathbb R^d$, put $p_i=(1,a_i)^{\mathsf T}$, $F=\mathbb Q((a_i)_\alpha)$, $B=F\otimes_{\mathbb Q}F$, and $m_F(x\otimes y)=xy$. The Ramsey quantifiers are $\forall r\in\mathbb Z_{\ge2}\ \exists D\in\mathbb Z_{\ge1}\ \forall c:\mathbb R^D\to\{1,\ldots,r\}$: some isometric embedding $\iota:A\to\mathbb R^D$ has $c\circ\iota$ constant. No measurability of $c$ or rescaling of $A$ is permitted. The classification claims this is equivalent to existence of $P\in\operatorname{Mat}_{d+1}(B)$ satisfying \[(p_i\otimes1)^{\mathsf T}P(1\otimes p_i)=0\quad(1\le i\le s),\qquad m_F(P_{\alpha\beta})=\delta_{\alpha\beta}\quad(1\le\alpha,\beta\le d).\] The constant-coordinate row and column have no additional multiplied-entry conditions. Corollaries say every set of at most five circle points is Ramsey, and for transcendental $a\in(-1,1)$ the cyclic kite $\{(-1,0),(1,0),(a,\sqrt{1-a^2}),(a,-\sqrt{1-a^2})\}$ is Ramsey while embedding in no finite transitive Euclidean set.

Outline of the argument

Finite Euclidean Ramsey classification: Tensor cancellation is first realized by exact pairs of permutation constructions. A group-endpoint criterion and finite-string realization prepare the distinct weights needed for a Hales--Jewett line to have precisely unit distance scale. Further averaging and short corrective paths equalize endpoints, making the sufficient tensor condition usable under arbitrary colorings.

Formal verification

Family 172 describes Lean coverage for the tensor criterion and the stated spherical, subtransitive, and circular configuration consequences.

Finite Euclidean Ramsey classification: The manuscript links EuclideanRamsey.lean (declaration OAI.EuclideanRamsey.classification), EuclideanRamseyCircle.lean, EuclideanRamseyNine.lean, EuclideanRamseyQuadratic.lean, EuclideanRamseySpherical.lean, EuclideanRamseyTransitive.lean, GrahamSpherical.lean (declaration OAI.GrahamSpherical.full_main).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

Go to contribution →