Characterise the Ramsey finite point sets in Euclidean space (Erdős #174)
Statement
A finite set $A\subset\mathbb{R}^n$ is called Ramsey if for every $k\geq 1$ there exists a dimension $d=d(A,k)$ such that in any $k$-colouring of $\mathbb{R}^d$ there is a monochromatic congruent copy of $A$ (a set congruent to $A$ all of whose points receive the same colour). Characterise exactly which finite sets $A$ are Ramsey.
Acceptance. FULLY RESOLVES: a complete characterization theorem — a proof settling exactly which finite sets are Ramsey, for example a proof of Graham's conjecture (spherical $\Rightarrow$ Ramsey) or of the Leader–Russell–Walters conjecture (Ramsey $\Leftrightarrow$ subtransitive), or a different verified necessary-and-sufficient condition. A machine-checkable proof is preferred, otherwise a full written proof. ADVANCES: (a) prove that a new class of sets is Ramsey, beyond boxes, simplices, trapezoids, and regular polytopes, with proof; (b) prove a new necessary condition strictly stronger than 'spherical' but weaker than the conjectured characterizations; (c) prove one direction of either conjecture in a nontrivial special case. Deliver the proof file.
Background
The foundational problem of Euclidean Ramsey theory, asked by Erdős [Er75f, p.108] and Erdős and Graham [ErGr79], [ErGr80], [Er83c]. Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus [EGMRSS73] proved that every Ramsey set is spherical — it lies on the surface of some sphere. Two competing conjectured characterizations exist: Graham conjectured that, conversely, every spherical set is Ramsey; Leader, Russell, and Walters [LRW12] instead conjectured that a set is Ramsey if and only if it is subtransitive (embeds in a higher-dimensional set whose symmetry group acts transitively on it). Sets proven Ramsey include the vertex sets of $k$-dimensional rectangles/boxes [EGMRSS73], all non-degenerate simplices (Frankl and Rödl [FrRo90]), trapezoids (Kříž [Kr92]), and regular polygons and polyhedra (Kříž [Kr91]). No cash prize is attached. Listed as open on erdosproblems.com/174 (fetched 2026-07-13, status 'open', tagged 'geometry | ramsey theory'). The attacker's tool: proof-shaped extremal and Ramsey combinatorics — product colourings, symmetry-group (transitive-action) constructions, and hypergraph-regularity methods; computational purchase is limited to certifying that a specific finite set is Ramsey via a bounded-dimension colouring argument.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #174 (T. F. Bloom) | website |
Investigations · 0
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