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arxiv: 2605.20951 · v1 · pith:SWKW7APBnew · submitted 2026-05-20 · 🧮 math.CO

Corrigendum and Addendum to "Fra\"{i}ss\'{e}'s Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition''

Dragan Ma\v{s}ulovi\'c This is my paper

Pith reviewed 2026-05-21 03:51 UTC · model grok-4.3

classification 🧮 math.CO
keywords big Ramsey degreesgeneric structures2-dimensional partial ordermonomorphic decompositionrelational structurespolynomial growthreductsFraisse conjecture
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The pith

A correction shows the generic 2-dimensional partial order has finite big Ramsey degrees.

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

This note corrects the identification of a reduct of the generic permutation as the generic partial order. The structure is in fact the generic 2-dimensional partial order. The arguments from the original paper hold with this change and establish that the structure has finite big Ramsey degrees. The addendum characterizes the existence of finite big Ramsey degrees for all countable relational structures whose language contains a linear order and whose age has polynomial growth.

Core claim

The central claim is that the reduct of the generic permutation is the generic 2-dimensional partial order. The framework for structures admitting finite monomorphic decomposition therefore shows that this structure has finite big Ramsey degrees. The addendum combines this analysis with other results to characterize exactly when countable relational structures that include a linear order and have an age of polynomial growth possess finite big Ramsey degrees.

What carries the argument

The general framework for proving finite big Ramsey degrees for structures that admit a finite monomorphic decomposition.

If this is right

  • The framework applies directly to establish finite big Ramsey degrees for the generic 2-dimensional partial order.
  • The same approach works for a previously unexplored class of generic relational structures.
  • The addendum supplies a complete characterization of finite big Ramsey degrees based on the presence of a linear order and polynomial growth of the age.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework may apply to other dimensions of generic partial orders in the same manner.
  • Re-examining similar reduct identifications in the literature could uncover further applications.
  • The characterization suggests examining structures with linear orders but different age growth rates as a boundary test.

Load-bearing premise

The arguments presented in Section 6 remain valid once the reduct is correctly identified as the generic 2-dimensional partial order.

What would settle it

Observation of a countable relational structure whose language contains a linear order, whose age has polynomial growth, and that lacks finite big Ramsey degrees would falsify the characterization given in the addendum.

Figures

Figures reproduced from arXiv: 2605.20951 by Dragan Ma\v{s}ulovi\'c.

Figure 1
Figure 1. Figure 1: A counterexample for the amalgamation property for the class of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (A × B, ⊑A×B) in the proof of Lemma A Let us show that there is exactly one way to reconstruct ⩽ A×B 1 and ⩽ A×B 2 from ⩽ A×B0 1 and ⩽ A×B0 2 . Let us start with an. Since an−1 ⊑A×B an it follows that an−1 ⩽ A×B 1 an and an−1 ⩽ A×B 2 an. By the induction hypothesis we have that b1 ⩽ A×B0 1 an−1, so b1 ⩽ A×B 1 an. Then an ⩽ A×B 2 b1 because an and b1 are incomparable with respect to ⊑A×B = (⩽ A×B 1 ) ∩ (⩽ A… view at source ↗
read the original abstract

In Section 6 of the paper ``Fra\"{\i}ss\'{e}'s Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition'', we applied the methods developed in earlier sections to show that a certain reduct of the generic permutation has finite big Ramsey degrees. Unfortunately, this reduct was incorrectly identified as the generic partial order. We are grateful to Jan Hubi\v{c}ka for bringing this error to our attention. In this note we correct the statements that rely on this misidentification and demonstrate that the reduct in question is in fact the generic 2-dimensional partial order. We emphasize that the arguments presented in Section 6 remain valid, with the sole exception of the Claim in the proof of Theorem 6.4, whose role was to (incorrectly) identify the reduct of the generic permutation as the generic partial order. This correction has an unexpected positive consequence. Rather than reproving a well-known result whose existing proof is already notably elegant, this note demonstrates that our general framework can be used to establish that a previously unexplored class of generic relational structures has finite big Ramsey degrees. This observation opens a potentially new direction for further research in the thriving area of big Ramsey combinatorics. In the addendum, we combine a recent result by Oudrar and Pouzet with our analysis of finite big Ramsey degrees for structures admitting finite monomorphic decomposition to characterize the existence of finite Big Ramsey degrees for all countable relational structures whose language has a linear order and age has polynomial growth.

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

1 major / 0 minor

Summary. This corrigendum corrects an identification error from Section 6 of the authors' prior paper: a reduct of the generic permutation was misidentified as the generic partial order but is in fact the generic 2-dimensional partial order. The note asserts that all arguments in Section 6 carry over unchanged except for the Claim in the proof of Theorem 6.4, whose sole role was the incorrect identification. It notes a positive consequence—that the monomorphic-decomposition framework now applies to this previously unexplored class—and, in the addendum, combines the authors' analysis with the Oudrar–Pouzet result to characterize finite big Ramsey degrees for all countable relational structures whose language contains a linear order and whose age has polynomial growth.

Significance. If the validity assertion holds, the note both rectifies the record and shows that the general framework yields finite big Ramsey degrees for a new family of generic structures, opening a research direction in big Ramsey combinatorics. The addendum supplies a clean, broadly applicable characterization that integrates monomorphic decomposition with an external polynomial-growth result.

major comments (1)
  1. Corrigendum, paragraph beginning 'We emphasize that the arguments presented in Section 6 remain valid': the assertion that the misidentification affects only the Claim in the proof of Theorem 6.4 is stated without any sketch of the unaffected steps or why the reduct identification error does not propagate. Because this assertion is load-bearing for the claimed positive consequence (applicability to the generic 2-dimensional partial order), a short outline of the preserved arguments would make the claim verifiable within the note itself.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the corrigendum and for the constructive suggestion to improve verifiability. We will revise the note to incorporate a brief outline addressing the concern.

read point-by-point responses

  1. Referee: Corrigendum, paragraph beginning 'We emphasize that the arguments presented in Section 6 remain valid': the assertion that the misidentification affects only the Claim in the proof of Theorem 6.4 is stated without any sketch of the unaffected steps or why the reduct identification error does not propagate. Because this assertion is load-bearing for the claimed positive consequence (applicability to the generic 2-dimensional partial order), a short outline of the preserved arguments would make the claim verifiable within the note itself.

    Authors: We agree that the current phrasing would benefit from additional clarification to make the claim self-contained. The identification error was isolated to the Claim, which served only to name the structure as the generic partial order; all other steps in Section 6 rely on the general theory of finite monomorphic decompositions developed in the earlier sections of the original paper, which apply directly once the correct structure (the generic 2-dimensional partial order) is substituted. In the revised version we will insert a short paragraph immediately after the emphasized statement that briefly lists the unaffected components: verification of the finite monomorphic decomposition property for the reduct, invocation of the main theorems on big Ramsey degrees for such structures, and the resulting conclusion that the generic 2-dimensional partial order has finite big Ramsey degrees. This addition will render the positive consequence verifiable without requiring the reader to consult the original paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

This corrigendum isolates the sole error to one Claim in the proof of Theorem 6.4 of the prior paper and states that all other arguments in Section 6 carry over unchanged to the generic 2-dimensional partial order. The addendum combines an external result of Oudrar and Pouzet with the existing monomorphic-decomposition framework; no derivation step, equation, or claim in the present note reduces by construction to a self-definition, fitted input, or unverified self-citation chain. The document is therefore self-contained as a correction note.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a short correction note whose central claims rest on the validity of arguments from the authors' prior paper and on a recent external result by Oudrar and Pouzet; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The arguments presented in Section 6 remain valid with the sole exception of the Claim in the proof of Theorem 6.4.
    Explicitly stated in the abstract as the basis for keeping the rest of the section intact.

pith-pipeline@v0.9.0 · 5823 in / 1189 out tokens · 29990 ms · 2026-05-21T03:51:24.316990+00:00 · methodology

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Reference graph

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