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Definitions (p. 7)

A template over $[m]$ is a non-decreasing word $T\in[m]^l$ for some $l$. Let $S$ be the set of words of length $l$ that are rearrangements of $T$. A block set with template $T$ in $[m]^n$ is formed by choosing pairwise disjoint sets $I_1,\ldots,I_l\subset[n]$, all of the same size $d$, and a letter $a_i$ for each position $i$ outside their union; it consists of the words $w\in[m]^n$ with $w_i=a_i$ outside the blocks and, for some $v\in S$, $w_i=v_j$ for every $i\in I_j$. The common size $d$ is called the block size or degree.

Statement

Conjecture 6 ([10]) (p. 7). "Let $m$ and $k$ be positive integers and let $T$ be a template over $[m]$. Then there exist positive integers $n$ and $d$ such that whenever $[m]^n$ is $k$-coloured there exist [sic] a monochromatic block set of degree $d$ with template $T$."

The paper attributes the conjecture to Leader, Russell and Walters, its reference [10] (Transitive sets in Euclidean Ramsey theory, J. Combin. Theory Ser. A 119 (2012), 382–396), where it first appears. It is posed, not proved, here. The corpus's record of that paper's own convention, in which "degree" counts all active positions $ld$ rather than the common block size $d$, is on the external-inputs page.

Source. M.-R. Ivan, I. Leader and M. Walters, Generalised Prisms and Euclidean Ramsey Theory, arXiv:2606.13472v1 (11 June 2026), Section 3, definitions and Conjecture 6, p. 7.

Read depth. Claims checked: the definitions and the conjecture were read clause by clause on the PDF.

Context in the paper

The paper (p. 7) records the conjecture as verified for the templates $1\,2^s\,3^t$ ($s,t\in\mathbb N$) in [10] and for $1234$ (from Kříž's theorem and the solubility of $S_4$), and notes that blocks of size one do not suffice for $123$, while blocks of size two do (its reference [6]). These are the paper's pointers, not results checked here.

To a template it attaches a geometric set: given reals $\alpha_1,\ldots,\alpha_n$ [sic; one value per letter of $[m]$ is meant], the points of $\mathbb R^l$ whose coordinates contain each $\alpha_i$ as many times as $T$ contains $i$. It states that the conjecture for $T$ makes this set Ramsey. The corpus's argument for that implication: given $k$, take the $n$ and $d$ of the conjecture, colour a word $w\in[m]^n$ by the colour of the point $d^{-1/2}(\alpha_{w_1},\ldots,\alpha_{w_n})$, and observe that the image of a monochromatic block set is a congruent copy of the geometric set, since each block contributes $d$ equal squared differences. Repeated or dependent values $\alpha_i$ do no harm.

The paper then states that for algebraically independent $\alpha_i$ the only symmetries of the geometric set are the coordinate permutations, and defines the symmetry group of a template to be $S_l$ acting on $S$ by permuting coordinates. The first assertion fails for some templates, for example $1122$; see the corpus's counterexample. The definition itself is what Theorem 7 uses.

Bears on

• Problem 174: the block sets conjecture implies that every subtransitive set is Ramsey (p. 7), which is one direction of the Leader–Russell–Walters conjecture, recalled on p. 1, that the Ramsey sets are exactly the subtransitive sets. The conjecture is stated, not proved, in this paper.