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Statement

Conjecture 8 (p. 9). "Let $X$ be a Ramsey set in $\mathbb R^{d}$ and let $z$ be a point in $\mathbb R^{d+1}$ that does not belong to the hyperplane containing $X$. Then the set $X\cup\{z\}$ is Ramsey."

It asks whether Corollary 4 extends to bases that are merely assumed Ramsey, whether or not they are subtransitive (p. 9).

Source. M.-R. Ivan, I. Leader and M. Walters, Generalised Prisms and Euclidean Ramsey Theory, arXiv:2606.13472v1 (11 June 2026), Section 4, Conjecture 8, p. 9; the section runs pp. 9–10.

Read depth. Claims checked: the conjecture and the remarks around it were read clause by clause on the PDF.

The paper's remarks (pp. 9–10)

• The conjecture would follow from either of the two competing conjectured descriptions of the Ramsey sets: that they are the spherical sets, or that they are the subtransitive sets. The corpus's reasons: a pyramid over a spherical base is spherical (an explicit centre lies on the line through the base's centre perpendicular to the base's hyperplane); and a pyramid over a subtransitive base is subtransitive by the corpus's extension of Corollary 4.
• For a Ramsey set $X$: if the perpendicular distance from $z$ to the plane of $X$ exceeds the circumradius of $X$, then $X\cup\{z\}$ is at least 2-Ramsey, by a sphere-growing argument the paper sketches; the corpus's account, with the points it fills in, is on the two-colour observation page.
• Two prism-type configurations in $\mathbb R^3$ are not covered by Theorem 1, and the paper asks whether they are Ramsey: a rectangle with the same rectangle above it rotated about its centre by an irrational multiple of $\pi$; and a rectangle with two points above it at one height, their midpoint not above the rectangle's centre, the vector from that midpoint to the rectangle's centre perpendicular to the segment joining the two points (which the paper says makes the set spherical), and the angle between the segment and the rectangle not a rational multiple of $\pi$. For the first, a square is a regular polygon and is covered at every angle by Corollary 5, so the question concerns non-square rectangles.

Later work

Two papers of August 2026 prove the conclusion of Conjecture 8 for every finite Ramsey base and every point outside its affine hull: Moore's Theorem 1.2 (arXiv:2608.09649v1) and, by a different method, Mirabi's Theorem 1.1 (arXiv:2608.11736v1); their standing is recorded on their own pages. Those results give Ramsey sets, not the subtransitive or subsoluble enclosures of Corollary 4.

Bears on

• Problem 174: a closure property of the class of Ramsey sets that the paper conjectures, and notes would follow from either conjectured description of that class.