Which finite shapes must appear in one color

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Family 172 · Combinatorics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

Choose a finite shape, perhaps the corners of a triangle or several points on a circle. Color every point of a Euclidean space using finitely many colors. If the dimension is high enough, must there be a copy of the shape whose points all have the same color?

The copy must preserve every distance at the original scale. Rotations and translations are allowed; enlarging the shape is not. The coloring can be completely arbitrary, with no continuity or measurability assumption.

A finite configuration with this property is called Euclidean Ramsey. The manuscript claims a classification of all such configurations by an algebraic condition on their coordinates. Among its consequences, every set of at most five points on a circle is Ramsey, while some larger spherical sets fail the condition. It also separates the Ramsey property from the ability to sit inside a finite set with transitive symmetry.

2. Background and essential definitions

A finite nonempty set $A$ is Ramsey if, for every integer $r\geq2$, there is a dimension $D$ such that every coloring

$$c:\mathbb R^D\longrightarrow\{1,\ldots,r\}$$

is constant on a congruent copy of $A$. The dimension may depend on both $A$ and $r$. Congruence means exact preservation of pairwise Euclidean distances.

A set is spherical if all its points lie on one sphere. A circle counts as a sphere in a plane. A finite Euclidean set is transitive if its distance-preserving symmetries can move any point to any other point. A regular polygon is an example. A set is subtransitive if it is congruent to a subset of some finite transitive set, possibly in a larger dimension.

To state the classification, place $A=\{a_1,\ldots,a_s\}$ in its affine span, identified with $\mathbb R^d$. The affine span is the smallest translated linear subspace containing it. Let $F$ be the smallest field containing the rational numbers and all coordinates of these points.

The tensor product $F\otimes_{\mathbb Q}F$ keeps two copies of this field algebraically separate. Its elements are finite sums of symbols $x\otimes y$, subject to addition and rational bilinearity. In particular,

$$(x+x')\otimes y=x\otimes y+x'\otimes y, \qquad (qx)\otimes y=x\otimes(qy)$$

for rational $q$. One may multiply symbols by multiplying their two coordinates separately.

There is a map $m_F$ that multiplies the two factors:

$$m_F(x\otimes y)=xy.$$

An expression can become zero after applying this map without being zero in the tensor product. Keeping that extra information is the point of the criterion. Ordinary equations describing a sphere see only the multiplied expressions.

3. Development history and previous partial results

Erdős, Graham, Montgomery, Rothschild, Spencer and Straus introduced Euclidean Ramsey theory in 1973. They proved that every finite Ramsey configuration is spherical. Graham later conjectured that this necessary condition might also be sufficient.

Positive results accumulated for geometrically important classes. Frankl and Rödl proved that nondegenerate triangles, and then all nondegenerate simplices, are Ramsey. A nondegenerate simplex consists of affinely independent vertices, such as a genuine triangle or tetrahedron. Their simplex theorem is a primary landmark. Kříž treated certain transitive symmetry groups and cyclic trapezoids, and Cantwell proved the result for regular polytopes.

A different possible classification emerged from symmetry. Leader, Russell and Walters proposed that Ramsey configurations are exactly the subtransitive ones. This would explain many known positive examples through a common geometric source. In a companion paper, they constructed cyclic quadrilaterals that are not subtransitive, making four-point circle configurations an important test.

The manuscript also places itself after several recent developments. Results on adjoining a point outside the affine span extended the known Ramsey classes, but did not settle arbitrary collections of coplanar circle points. Pálvölgyi’s September 2026 manuscript presents a seven-point circle configuration that is not Ramsey, disproving the spherical characterization. Its author also announced a proof for all finite transitive sets, with exposition still in preparation at the version cited here.

Against this background, the new manuscript claims an exact criterion rather than another isolated positive or negative family. Sphericity and transitive symmetry become consequences or sufficient conditions within a larger algebraic picture.

4. What the new manuscripts claim

The exact condition

A singleton is Ramsey. For a configuration of at least two distinct points with affine span $\mathbb R^d$, put

$$p_i=(1,a_i)^T, \qquad B=F\otimes_{\mathbb Q}F.$$

The initial coordinate one allows affine expressions to be written as matrix expressions. The manuscript states that $A$ is Ramsey if and only if there is a $(d+1)$-by-$(d+1)$ matrix $P$ with entries in $B$ satisfying

$$(p_i\otimes1)^T P(1\otimes p_i)=0 \qquad(1\leq i\leq s),$$

and

$$m_F(P_{\alpha\beta})=\delta_{\alpha\beta} \qquad(1\leq\alpha,\beta\leq d).$$

Here tensors are formed coordinate by coordinate, and $\delta_{\alpha\beta}$ is one when the indices agree and zero otherwise. The second condition says that multiplying the entries of the spatial block gives the identity matrix. It imposes no separate condition on the multiplied constant row or column.

The first equations must hold in the tensor ring itself. Applying $m_F$ to them produces an ordinary equation

$$\|a_i\|^2+\ell\cdot a_i+c=0$$

for suitable $\ell$ and $c$. Completing the square gives a sphere. Thus the classification retains the classical spherical necessity but adds a stronger algebraic requirement before multiplication.

Geometric consequences

The paper claims that every nonempty subtransitive set satisfies the criterion. It also gives a sufficient condition using quadratic interpolation: quadratic polynomials must be able to prescribe arbitrary values on the chosen points. For spherical configurations satisfying this independence condition, a suitable tensor matrix exists.

This applies to every nonempty set of at most five distinct points on a circle. In particular, the non-subtransitive cyclic quadrilaterals constructed by Leader, Russell and Walters are Ramsey. The necessity direction of their proposed characterization therefore fails, according to the manuscript.

The paper also supplies negative examples illustrating the criterion: nine circle points obtained from algebraically independent parameters, and a twelve-point configuration made from three rotated squares. The nine-point conclusion is not presented as the first spherical obstruction; the earlier seven-point result is explicitly credited.

5. Why the advance matters

Importance: 5/5. A necessary-and-sufficient characterization for an entire classical class is an exceptional advance. It changes the question from which special geometric shapes can be handled to what exact algebraic feature governs all finite configurations.

The result also clarifies two attractive but incomplete intuitions. Lying on a sphere is necessary, yet does not capture every obstruction. Having enough finite symmetry is sufficient, yet is not necessary. The tensor condition identifies information that survives between these two descriptions.

For readers outside Ramsey theory, the striking point is that an assertion about every possible coloring of a continuum is claimed to be equivalent to an exact finite-coordinate certificate. The certificate is algebraic rather than a simple visual rule, so classification does not automatically mean easy computation. Nevertheless, it offers a common language for both constructions and impossibility examples.

6. The main idea

Retain identities that ordinary multiplication erases

The proof first rewrites the matrix condition using tensors of affine functions. An affine function is a linear function plus a constant. Evaluating the tensor at each point gives zero, while its gradient data reproduce the Euclidean metric through an identity matrix.

If no such tensor exists, an algebraic separation argument constructs a finite coloring that avoids the configuration in every dimension. This is the necessity direction: failure of the certificate becomes an obstruction to monochromatic copies.

For sufficiency, the tensor identities are converted into finite lists of affine functions with signed rational multiplicities. At each point of $A$, positive and negative contributions cancel exactly by evaluation value. The paper uses these cancellations to produce two arrays of vectors representing copies at different scales, with matching coordinate multisets.

Those coordinate multisets can be rearranged by permutations. The construction must exploit the rearrangements while retaining exact distances; an approximation argument alone would not work for arbitrary colorings.

Synchronize the scale before asking for a color

The second part uses word and group methods. A word is a finite string over a finite alphabet. The Hales–Jewett theorem guarantees a monochromatic family of words obtained by changing a common set of variable positions together, once the word length is large enough.

When coordinates of Euclidean copies are placed in separate blocks, squared distances add. If a monochromatic word pattern has $t$ variable blocks, it can therefore multiply a squared distance by $t$. But the desired copy must have scale one, and $t$ is initially unknown.

The manuscript prepares compatible constructions with weights $1,1/2,\ldots,1/n$, where $n$ is the chosen word length. Whichever number $t$ of variable positions appears, the weight $1/t$ cancels that multiplier exactly. Additional group constructions align the endpoints associated with the coordinate permutations. Compactness then yields a finite ambient dimension for each prescribed number of colors.

This explains the two demanding parts of the argument: preserving exact algebraic cancellations and coordinating the scale selected by a combinatorial theorem. Neither can be replaced by saying that the resulting shape is merely close to the intended one.

7. Scope, limitations, and open questions

The classification is for finite nonempty configurations, arbitrary finite colorings, and congruent copies at the original scale. Allowing a free dilation would change the problem. Requiring measurable colorings would also change what kinds of obstruction are relevant.

The coordinate-field condition is invariant under replacing the configuration by a congruent representative. It depends on exact relations among coordinates, however. Decimal approximations to arbitrary real coordinates do not supply those relations, so the statement is not a numerical decision algorithm for a drawing or a list of rounded measurements.

The theorem guarantees some finite dimension for each configuration and number of colors. Its headline does not provide a practical optimal dimension bound. Finding usable certificates or obstructions for concrete families, and making the dimension dependence effective and sharp, remain separate tasks.

The five-point circle consequence and the existing seven-point obstruction also leave an evident small-configuration question: how the criterion behaves for six circle points. The classification supplies a framework for that investigation without turning every intermediate case into an immediate corollary.

8. Sources and further reading

The manuscript’s first section states the criterion and explains why ordinary sphere equations lose information. Its final section derives the subtransitive, circle-point and negative-example consequences.

For primary background, see Frankl–Rödl on simplices, Leader–Russell–Walters on transitive sets, their cyclic quadrilateral examples, and Pálvölgyi’s non-Ramsey heptagon. The bibliography of M1 gives the original Euclidean Ramsey papers and the word-theoretic sources behind the exact-scale construction.

Source manuscripts

• M1. OpenAI, A classification of finite Euclidean Ramsey configurations (September 23, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.