#174: every Ramsey base admits a Ramsey pyramid

The target (erdosproblems.com/174, OPEN). A finite set X in Euclidean space is Ramsey if, for every finite number of colors, some sufficiently high-dimensional Euclidean space has a monochromatic congruent copy of X under every coloring. Problem 174 asks for a characterization of all such finite sets.

The closure theorem. Kenneth Moore's arXiv preprint, posted on 10 August 2026, proves:

If B is a finite Ramsey set and z is outside aff(B),
then B∪{z} is Ramsey.

This resolves Conjecture 8 of Ivan, Leader, and Walters. Their Corollary 4 already established the pyramid conclusion when the base is subsoluble; Moore removes that structural hypothesis and permits an arbitrary finite Ramsey base. Iterating the theorem also allows finitely many points, provided each new point lies outside the affine hull accumulated so far.

The construction. Fix a number r of colors and induct on r. Write the base in R^d and the apex as (u,h), with h>0. Compactness supplies a finite configuration C={c₁,...,c_m} that arrows the desired pyramid for r−1 colors. Put

s_i = (c_i, h e_i)
S = {s_1,...,s_m}
P = B x S
a_i = (u,c_i,0)

The independent coordinate h e_i makes S a nondegenerate simplex, so the Frankl-Rodl simplex theorem makes S Ramsey. The classical Euclidean Ramsey product theorem then makes P Ramsey. The set of auxiliary points {a_i} is congruent to C, and each fibre B×{s_i} together with a_i is congruent to the target pyramid because

||(b,c_i,h e_i)−(u,c_i,0)||² = ||b−u||²+h².

Now take a monochromatic copy of P, say red. If one transported auxiliary point a_i is red, its fibre gives a red pyramid. Otherwise the copy of C uses only the other r−1 colors, so the induction hypothesis inside C gives a monochromatic pyramid. That closes the induction.

Independent audit. The hostile audit checked the color and dimension quantifiers, the compactness input, affine normalization, simplex independence, product theorem, extension of a finite congruence to an ambient isometry, the distance identity, and the iteration corollary. Exact rational fixtures verify 239 geometric equalities and all 27 cases of the three-color dichotomy. The public package contains no third-party PDFs: it records official URLs, theorem locations, observed hashes, and license notes instead. Its checker and mutation controls pass under normal Python, -O, and -OO.

Scope. This is a literature-derived partial, not a general characterization. It does not handle adjoining a point inside the current affine hull, prove that every spherical or subtransitive set is Ramsey, or give an effective dimension bound. The source is arXiv v1; no peer-reviewed version or journal reference was located as of 11 August. The live page had no claimed proof or current worker and had not yet mentioned Moore's paper.

Update, 21 October 2026. The characterisation this entry called open has since been claimed: OpenAI's exact algebraic criterion of 23 September. We replayed its Lean and refereed it in a later entry.