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arxiv: 2406.01459 · v1 · submitted 2024-06-03 · 🧮 math.CO

Block Sizes in the Block Sets Conjecture

Maria-Romina Ivan , Imre Leader , Mark Walters This is my paper

Pith reviewed 2026-05-24 00:02 UTC · model grok-4.3

classification 🧮 math.CO
keywords block sets conjectureEuclidean Ramseycombinatorial conjecturetemplatesblock sizesRamsey theoryHales-Jewett theorem
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The pith

The sizes of the blocks required in the block sets conjecture cannot be bounded by any constant, even for templates over a three-symbol alphabet.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the block sizes needed to satisfy the block sets conjecture grow without bound as one considers different templates, at least when those templates are built from an alphabet of three symbols. For the particular template 123 the paper proves that blocks of size two are always enough, no matter how many colours are involved, and that this size is the smallest possible. A reader would care because the block sets conjecture is a combinatorial stand-in for the open question of which geometric configurations are guaranteed to appear monochromatically in any colouring of high-dimensional space. If the block sizes must increase without limit, then any proof of the conjecture would have to accommodate this growth rather than rely on a uniform block size.

Core claim

The paper proves that there is no fixed bound on the size of blocks that works for all templates with three symbols in the block sets conjecture. Separately, it shows that when the template is the sequence 123, blocks of size two suffice for colourings with any number of colours, and that no smaller size works.

What carries the argument

The block sets conjecture on blocks in large products over a template, with explicit constructions forcing larger blocks for three-symbol cases and direct verification that size two suffices for the 123 template.

Load-bearing premise

The combinatorial definitions and reduction from the Euclidean Ramsey question to the block sets conjecture correctly capture the transitive case.

What would settle it

A single number N such that blocks of size at most N suffice for every three-symbol template would falsify the claim that sizes cannot be bounded.

read the original abstract

A set $X$ is called Euclidean Ramsey if, for any $k$ and sufficiently large $n$, every $k$-colouring of $\mathbb{R}^n$ contains a monochromatic congruent copy of $X$. This notion was introduced by Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus. They asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. It is not too difficult to show that if a set is not spherical then it is not Euclidean Ramsey either, but the converse is very much open despite extensive research over the years. On the other hand, the block sets conjecture is a purely combinatorial, Hales-Jewett type of statement, concerning `blocks in large products', introduced by Leader, Russell and Walters. If true, the block sets conjecture would imply that every transitive set (a set whose symmetry group acts transitively) is Euclidean Ramsey. As for the question above, the block sets conjecture remains very elusive, being known only in a few cases. In this paper we show that the sizes of the blocks in the block sets conjecture cannot be bounded, even for templates over the alphabet of size $3$. We also show that for the first non-trivial template, namely $123$, the blocks may be taken to be of size $2$ (for any number of colours). This is best possible; all previous bounds were `tower-type' large.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves two results on the block sets conjecture of Leader-Russell-Walters: that block sizes cannot be bounded for any template over a 3-letter alphabet, and that for the specific template 123 the blocks may be taken of size 2 for every number of colours (and that this is optimal).

Significance. If the proofs are correct, the first result shows that no uniform bound on block size exists even in the smallest non-trivial alphabet, while the second replaces previous tower-type bounds with a tight constant for the 123 template. Both statements are direct combinatorial claims about the conjecture itself and therefore bear on the question of whether transitive sets are Euclidean Ramsey.

minor comments (2)
  1. [Introduction] The introduction should include a self-contained statement of the block sets conjecture (including the precise definition of a template and of a block) rather than relying solely on the citation to Leader-Russell-Walters.
  2. [Abstract] The claim that 'all previous bounds were tower-type large' would be strengthened by a brief citation or footnote indicating which earlier results are being referenced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our results and for recommending minor revision. The report accurately captures the two main contributions: unbounded block sizes over a 3-letter alphabet, and the tight bound of block size 2 for the 123 template.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper advances two direct combinatorial results on the externally introduced block sets conjecture: unbounded block sizes for any 3-letter template, and an explicit size-2 construction for the 123 template. These are established via explicit constructions and arguments that stand independently of the cited prior reduction; the self-citation to Leader-Russell-Walters merely identifies the conjecture and is not used to justify or derive the new bounds. No equation reduces a claimed prediction to a fitted input, no ansatz is smuggled, and no uniqueness theorem is invoked from overlapping authors. The derivation chain is therefore self-contained against the stated combinatorial definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard definitions and framework of the block sets conjecture from prior literature; no free parameters, invented entities, or ad-hoc axioms are introduced in the stated claims.

axioms (1)
  • domain assumption Definitions of blocks, templates, and the block sets conjecture as introduced by Leader, Russell and Walters.
    The paper builds directly on this prior combinatorial framework to state its results.

pith-pipeline@v0.9.0 · 5795 in / 1178 out tokens · 21243 ms · 2026-05-24T00:02:52.720772+00:00 · methodology

discussion (0)

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Reference graph

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9 extracted references · 9 canonical work pages

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