Results on Euclidean Ramsey sets: the classification, the tools of its two halves, and examples on circles, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Classification of finite Euclidean Ramsey sets (OpenAI) — Theorem. A set A of s >= 2 points with affine span R^d is Ramsey if and only if some matrix P over B = F (x)_Q F, F the coordinate field, satisfies (p_i (x) 1)^T P (1 (x) p_i) = 0 for all i and m_F(P_ab) = delta_ab on the spatial block. Proved in the eighth lesson.
• Matrix condition and tensor condition — Proposition 2.1. The matrix condition holds if and only if some symmetric tensor of affine functions over Q has (e_i (x) e_i)(T) = 0 at every point and gradient Gram matrix I_d.
• The condition forces a sphere — Proposition 3.1. A set satisfying the matrix condition lies on a sphere.
• Colouring obstruction — Proposition 2.1. If the tensor condition fails for A, then (2s+1)^s colours suffice to colour every R^D without a monochromatic congruent copy of A. The colouring uses Q-linear functionals chosen with Zorn's lemma.
• Ramsey sets are spherical (Erdős, Graham, Montgomery, Rothschild, Spencer, Straus) — Corollary 2.2. Every Ramsey set lies on a sphere. No hypothesis.
• Finite lattice identity — Lemma 2.1. If a symmetric rational matrix C vanishes in every (Q^k/V_i) (x) (Q^k/V_i), there are finitely supported rational weights on Z^k with zero pushforward to every Z^k/Lambda_i, zero total weight and first moment, and second moment C.
• Small negative curvature — Lemma 1.1. If UCU^T = I_d, then for every eta > 0 some nonnegative smooth b with compact support has integral of (D_C b)_- |Ux|^2 below eta times the integral of b.
• Two-scale configurations — Theorem 3.1. Under the tensor condition, for every eta > 0 there are a congruent copy (f_i) of A and a copy (g_i) scaled by sqrt(q), q = eta/(1+eta), with each g_i a coordinate permutation of f_i.
• Hales–Jewett theorem — Theorem 1.1. For all k, r there is n such that every r-colouring of [k]^n has a monochromatic combinatorial line. No hypothesis.
• Ultrafilter space — Lemma 1.2. The space of ultrafilters on a set is compact and Hausdorff. No hypothesis.
• The corner group — Proposition 2.2. For a monoid S there is an idempotent p in beta S such that p (beta S) p is a group with identity p.
• Simultaneous realization — Lemma 3.1. If B belongs to the products q_i1 ... q_iM of ultrafilters on strings for every row i, with columns common or principal, then strings, equal across rows in common columns, have their row products in B.
• Path criterion — Proposition 1.1. If for every n two paths in K^s have the same endpoint and weights lambda > 0, mu >= 0 with mu/lambda < 1/n, then A is Ramsey.
• A finite dimension — Lemma 2.1. If every r-colouring of the space of finite sequences is constant on some congruent copy of A, then so is every r-colouring of some R^D.
• Pair projections are groups — Lemma 1.1. For i != j the projection of the correction monoid to the coordinates i, j is a group containing N x N, N the kernel of the exponent sum.
• Equal endpoints — Proposition 3.1. Two-scale configurations with ratio q give, for every delta > 0, two paths with a common endpoint and weight ratio at most q + s delta; with the path criterion this proves the sufficiency in the classification.
• Subsets of transitive sets — Corollary 1.1. Every nonempty subset of a finite transitive set is Ramsey. No hypothesis.
• At most five points on a circle — Corollary 2.2. Every nonempty set of at most five points on a circle is Ramsey; in particular every cyclic quadrilateral and every triangle.
• Kites outside transitive sets (Leader, Russell, Walters) — Theorem 3.2. Four distinct concyclic points with w = z + alpha(x - z) + beta(y - z), alpha != 1 and beta transcendental over Q(alpha), lie in no finite transitive set.
• Ramsey kites that are not subtransitive — Corollary 3.3. For transcendental a in (-1, 1) the kite K_a is Ramsey and not subtransitive.
• Nine generic points on a circle — Proposition 4.1. Nine points of the unit circle with algebraically independent parameters do not form a Ramsey set.
• Three rotated squares — Proposition 5.1. The twelve vertices of three squares on the unit circle, rotated by 0 and by plus or minus 2 arctan t with t a Liouville number, do not form a Ramsey set.

Reading order

Read the first lesson first: it states the classification and the tensor condition used throughout. The second lesson proves that the condition is necessary; the third to eighth lessons prove that it suffices, and the fifth and sixth lessons can be read on their own. The last lesson applies the classification to sets on circles.