Block sizes in the block sets conjecture

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definitions: Fixes the positive block-size convention and the order pattern used in the two block-size theorems, distinguishing the older total-degree convention.

external_inputs: States the finite Ramsey, line, and geometric inputs used in the complete relative deductions without importing unproved conjecture implications.

fixed_pattern: Proves the finite product-coloring argument that fixes the block pattern before the number of colors whenever the block size can be fixed.

generalized_obstruction: Proves the three-letter obstruction with p at least d and records why the source's unrestricted parameter claim is false.

geometric_power_scope: A histogram coloring of every regular hexagon power disproves the source's asserted Ramsey property at the fixed contraction factor one over root two.

geometric_scale_obstruction: Proves the geometric consequence of the block-size obstruction with an explicit squared-scale conversion and a repeated-letter rigidity input.

norm_observations: Expands the finite vector, residue-coloring, and metric observations while preserving the distinction between an arithmetic progression and distances.

open_questions: Records the paper's unresolved statements with positive parameters and separates their quantifiers from the proved block-size results.

theorem_2: Gives an explicit finite coloring that excludes all nonempty blocks of size at most d for the template consisting of one 1, d twos, and d cubed threes.

theorem_3: Reconstructs the finite Ramsey reduction and all six word substitutions giving an optimal degree-two block set for every number of colors.

three_letter_rigidity: Extends the three-distinct-letter rigidity lemma to positive multiplicities, supplying the precise extraction needed for the geometric scale obstruction.

two_singleton_symbols: Expands the source's generalization using a palindromic pattern with q plus two blocks, each of size two.

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Maria-Romina Ivan, Imre Leader, and Mark Walters, Block sizes in the block sets conjecture, Forum of Mathematics, Sigma 14 (2026), e67, 1–9, DOI 10.1017/fms.2026.10212. The paper was received 4 June 2024, accepted 21 February 2026, and published online 27 April 2026. The selected published PDF was obtained from the University of Cambridge repository and carries the publisher's journal header and CC BY 4.0 notice. Its physical and printed page numbers both run from 1 to 9.

The arXiv v1 PDF, also read for this card but not held, is arXiv:2406.01459v1, submitted 3 June 2024 at 15:48:59 UTC, with ten physical pages numbered 1–10. The arXiv record links the 2026 publication but records only this v1 manuscript. The source record identifies the exact artifacts and the principal version differences; it is not a claim that the PDFs are byte-identical. The published PDF (ivan_2026_block_sizes_block_sets_conjecture.pdf) prints on its first page "© The Author(s), 2026. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.", the Creative Commons Attribution 4.0 license. The arXiv record names arXiv's non-exclusive distribution license for the arXiv v1 PDF (arXiv:2406.01459), every other right reserved.

Main results and complete arguments

• Theorem 2 gives, for every $d\ge1$, one finite coloring of all three-letter words excluding template $1\,2^d3^{d^3}$ whenever each nonempty block has size at most $d$. It uses at most $(d+1)^{d^2+1}$ colors. Thus block size cannot be bounded independently of the template, even on three letters.
• Theorem 3 proves that degree two suffices for $123$, for every number of colors, with the fixed pattern $ABCCBA$. Degree one is excluded by Theorem 2. Its [[discrete_geometry/ivan_2026_block_sizes_block_sets_conjecture/two_singleton_symbols|extension to $12\,3^q$]] includes the complete palindromic construction for every $q\ge1$.
• The corrected generalization proves the lower bound for $1\,2^p3^q$ when $p\ge d$ and $q\ge p^2d$. The source omits the essential condition $p\ge d$.
• The geometric scale obstruction gives transitive Ramsey sets for which successful scaled-product witnesses must use an arbitrarily large expansion factor. Its repeated-letter rigidity proof expands a step not supplied in this paper, relative to the exact three-distinct-letter lemma of Leader–Russell–Walters.
• The finite-pattern argument and five lattice and norm deductions are complete at their stated elementary or external-input scopes.

The common block size is called degree here. In the older LRW source, degree instead means the total number of active coordinates. See the definitions and the exact input inventory. Finite Ramsey and Hales–Jewett remain explicit external theorems. The binary-template proof and template substitution already have canonical LRW pages and are linked rather than counted again.

Corrections and limits

Both versions print $d-1$ twice inside Theorem 2's proof although the argument proves the claimed bound $d$. Both omit $p\ge d$ in the following generalization; without it the statement conflicts with the degree-two $12\,3^q$ theorem. In Theorem 3's proof both list the seven-letter word $1221211$ among the balanced words of length $2k+2=6$. The reconstructed proofs identify these points and prove the stated repairs.

The source's product claim is false already for the regular hexagon at the asserted contraction factor $\sqrt2$. The complete histogram counterexample uses $3^6$ colors on every product dimension and proves the required affine and coordinate rigidity directly. It also refutes the more general $S_3$-transitive hexagon assertion. The published change from $60^\circ$ to $120^\circ$ repairs the angle description, not the product claim. None of these claims is an input to Theorems 2 or 3.

Additional notes correct the signed-support count in the discussion and two bibliographic entries. Corrections proved in this compilation are distinguished from the changes between the two source versions; no author-issued erratum is asserted.

The conjectures and questions retain the paper's exact quantifier distinctions. The main theorems do not prove the full block-sets conjecture, establish its fixed-degree strengthening for every template, or resolve #174. Historical claims cited from other papers and the speculative discussion of the $\ell_1$ norm are not counted as new complete proofs. This unit makes no formalization or local proof-assistant verification claim.

Only the edition under an open license is held; the source's other editions are not, since no license on record permits their redistribution, and the card cites the edition it names above.

Bears on

• E0174: the paper studies the block sets conjecture of Leader, Russell and Walters, which, if true, implies that every transitive set is Euclidean Ramsey (p. 2). Theorem 2 (p. 3) shows that no block size bound holds for all templates over $[3]$, and Theorem 3 (p. 5) shows that blocks of size $2$ suffice for template $123$ for every number of colors. The remark on pp. 4–5 gives transitive Ramsey sets that need an arbitrarily large scale factor in the product formulation. None of these results characterizes the Ramsey sets or proves the block sets conjecture; the paper's open conjectures and questions are recorded as posed.