The regular pentagon is canonically Ramsey

Abstract: A set of points $C\subset \mathbb{R}<sup>n$ is canonically Ramsey if there is some larger set of points $S\subset \mathbb{R}<sup>{n&#39;}$ such that any colouring of $S$ contains either a monochromatic copy of $C$ or a rainbow copy of $C$. Mao, Ozeki, and Wang introduced this notion, showing that the 30-60-90 triangle is canonically Ramsey. Since then, many other configurations have been shown to be canonically Ramsey. The author showed that cuboids are canonically Ramsey. Ge, Shu, Xu, and Yu later showed that all simplices are canonically Ramsey, after which the author showed that all products of simplices are canonically Ramsey, a class which, together with its closure under taking subsets, includes all previously known canonically Ramsey sets. We prove that regular polygons with a prime number of sides are canonically Ramsey---the first known sets outside this class.

• The paper proves that every regular polygon with a prime number of sides is canonically Ramsey, extending canonical Euclidean Ramsey theory beyond products of simplices.
• The proof constructs invariant copies through Ramsey-theoretic induction and uses modular arithmetic to show that any non-rainbow colouring forces a monochromatic copy.
• The result is stronger for powers of prime-sided polygons, while colourings of powers of composite polygons block the method and leave the regular hexagon as a central open case.

Context and main result

This paper, by Benedict Randall Shaw, establishes that regular polygons with a prime number of sides are canonically Ramsey. A configuration $C \subset \mathbb{R}^n$ is canonically Ramsey if there exists some configuration $S$ such that every colouring of $S$ contains either a monochromatic copy of $C$ or a rainbow copy of $C$. Canonical Euclidean Ramsey theory was introduced by Mao, Ozeki, and Wang (Mao et al., 2022), who showed the 30-60-90 triangle is canonically Ramsey; subsequent work established this property for squares and various triangles (Fang et al., 13 Oct 2025), hypercubes [10.5070/c65465673], all cuboids (Shaw, 2 Mar 2026), all simplices (Ge et al., 13 Jul 2026), and finally all products of simplices (Shaw, 16 Jul 2026)—a class containing every previously known canonically Ramsey set.

The paper's contribution is the first construction of canonically Ramsey sets outside the products-of-simplices class:

Theorem 1. For any prime $p$, the regular $p$-gon is canonically Ramsey.

In fact a stronger statement is proved:

Theorem 2. For any prime $p$, positive integer $k$, and regular $p$-gon $S$0, there is some $S$1 such that $S$2: any colouring of $S$3 contains a monochromatic or rainbow copy of $S$4.

This is genuinely stronger than Theorem 1: since it is unknown whether products of canonically Ramsey sets are canonically Ramsey, Theorem 2 cannot be obtained by iterating Theorem 1. The author notes an interesting consequence in the other direction—if products of canonically Ramsey sets were always canonically Ramsey, then $S$5 would fail to be canonically Ramsey, and any witnessing colouring could be shown to be neither spherical nor coordinate-permutation-invariant.

Proof strategy

Colourings are treated as equivalence relations $S$6 on the point set $S$7 (identifying the regular $S$8-gon with $S$9). A key device is the "local fragment" $S$0: for a word $S$1 with $S$2 wildcards, $S$3 is the pullback of $S$4 along the injection filling the wildcards. Words of dimension $S$5 thus induce equivalence relations on $S$6.

A scaled copy of $S$7 is called $S$8-invariant if each induced relation $S$9 depends only on the dimension of $C$0. Two auxiliary notions are introduced: $C$1 is $C$2-interchangeable if changing entries of $C$3 within $C$4 does not alter $C$5, and $C$6-swappable if swapping adjacent entries (at least one in $C$7) does not alter $C$8. Since $C$9-interchangeable plus $C$0-swappable implies $C$1-invariance, the proof reduces to building up these properties.

Step one: constructing a $C$2-invariant copy. Two lemmas drive an induction modelled on Kříž's argument for classical Ramsey sets [kriz1991]. First, via a finite Ramsey-theorem argument on $C$3-subsets coloured by their induced equivalence relations, any $C$4-interchangeable relation on a sufficiently large $C$5 restricts to a copy of $C$6 that is both $C$7-interchangeable and $C$8-swappable. Second, using the "standard emulated copy" embedding $C$9 of $p$0 into $p$1—which replaces each coordinate $p$2 by the full cyclic block $p$3—any $p$4-swappable relation becomes $p$5-interchangeable on the emulated copy. Inducting from the trivial base case ($p$6-swappability holds everywhere) yields a $p$7-invariant copy at scaling $p$8.

Step two: extracting monochromatic or rainbow structure. Within a $p$9-invariant copy, two structural lemmas are proved. If two points are equivalent, then for each coordinate $p$0, the transposed pair satisfies $p$1. Moreover, this commutation is transitive: $p$2 and $p$3 imply $p$4 (for dimension at least 4).

Now consider the standard emulated copy of $p$5 inside the invariant copy. If no rainbow copy exists, two distinct points $p$6 of this copy share a colour. Applying the first lemma across the cyclic blocks gives $p$7 for some nonzero difference $p$8 modulo $p$9, and iterating via transitivity extends this to $p$0 for all $p$1. Here primality is essential: since $p$2 generates the additive group mod $p$3, one obtains $p$4 for all pairs. Consequently every coordinate permutation preserves colour, so the entire emulated copy—which consists of points with identical coordinate multisets—is monochromatic. For composite $p$5, the multiples of $p$6 form a proper subgroup, and the argument collapses.

Limitations and open questions

The paper is explicit that the method does not extend to composite polygons, and identifies where the obstruction lies. For composite $p$7, the failure is not merely technical: the author constructs colourings of every $p$8 admitting no monochromatic or rainbow scaled copy of $p$9. For the hexagon, colour two points identically iff in each coordinate they agree or are opposite points; every scaled copy of $k$0 then receives exactly three colours. Analogous constructions exist for all composite $k$1. Moreover, the proof can be adapted to show that within large $k$2 one finds copies of $k$3 whose induced colouring has the same "difference divisible by fixed $k$4" form—so these obstructions are canonical under the proof's own machinery. The consequence is sharp: any witness to canonicity of a composite regular polygon cannot be a power $k$5, so a different ambient configuration would be required. The paper poses the natural next question: is the regular hexagon canonically Ramsey?

A second open question noted in the introduction is whether products of canonically Ramsey sets are canonically Ramsey; an affirmative answer would imply every canonically Ramsey set embeds in arbitrarily large ones and hence is itself Ramsey.

Conclusion

The paper proves that regular $k$6-gons, and all their powers, are canonically Ramsey for prime $k$7, providing the first canonically Ramsey sets beyond products of simplices. The proof combines a Kříž-style induction producing $k$8-invariant subconfigurations with an algebraic exploitation of primality, whereby a single coincident pair forces universal coordinate-transposition invariance and hence a monochromatic copy. The primality dependence is shown to be intrinsic to powers-of-polygons witnesses, leaving the composite case—and specifically the hexagon—as a concrete open problem.