Homogeneous Dual Ramsey Theorem
Pith reviewed 2026-05-25 02:31 UTC · model grok-4.3
The pith
For k dividing m there exists n multiple of m such that any N-coloring of homogeneous k-partitions of n has a homogeneous m-partition whose coarser k-partitions are monochromatic.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The homogeneous Dual Ramsey Theorem asserts that for positive integers k < m with k dividing m and any N, there exists n > m with m dividing n such that for every N-coloring of (n)^k_hom there is u in (n)^m_hom with all coarser t in (n)^k_hom monochromatic. The argument reduces the homogeneous case to the classical Dual Ramsey Theorem via an inductive construction.
What carries the argument
The sets (n)^k_hom of homogeneous k-partitions, meaning partitions of {1,...,n} into k classes of equal cardinality, together with the coarser-than relation between such partitions.
Load-bearing premise
An inductive construction exists that reduces any coloring problem on homogeneous partitions to an instance of the classical non-homogeneous Dual Ramsey Theorem.
What would settle it
Specific integers k < m with k dividing m and some N for which every candidate n multiple of m admits an N-coloring of (n)^k_hom with no u in (n)^m_hom whose coarser homogeneous k-partitions are all one color.
read the original abstract
For positive integers $k < n$ such that $k$ divides $n$, let $(n)^k_{\hom}$ be the set of homogeneous $k$-partitions of $\{1, \dots, n\}$, that is, the set of partitions of $\{1, \dots, n\}$ into $k$ classes of the same cardinality. In the article "Ramsey properties of infinite measure algebras and topological dynamics of the group of measure preserving automorphisms: some results and an open problem" by Kechris, Sokic, and Todorcevic, the following question was asked: Is it true that given positive integers $k < m$ and $N$ such that $k$ divides $m$, there exists a number $n>m$ such that $m$ divides $n$, satisfying that for every coloring $(n)^k_{\hom}=C_1\cup\dots\cup C_N$ we can choose $u\in (n)^m_{\hom}$ such that $\{t\in (n)^k_{\hom}: t\mbox{ is coarser than } u\}\subseteq C_i$ for some $i$? In this note we give a positive answer to that question. This result turns out to be a homogeneous version of the finite Dual Ramsey Theorem of Graham-Rothschild. As explained by Kechris, Sokic, and Todorcevic in their article, our result also proves that the class $\mathcal{OMBA}_{\mathbb Q_2}$ of naturally ordered finite measure algebras with measure taking values in the dyadic rationals has the Ramsey property.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper answers positively the open question of Kechris-Sokic-Todorcevic by proving that for positive integers k<m with k dividing m and any N, there exists n>m with m dividing n such that every N-coloring of the homogeneous k-partitions of [n] admits a homogeneous m-partition whose coarser homogeneous k-partitions are monochromatic. The proof is obtained by an explicit reduction to the classical (non-homogeneous) finite Dual Ramsey theorem of Graham-Rothschild: a sufficiently large multiple n is chosen and the given homogeneous coloring is canonically extended so that monochromaticity on the relevant coarser partitions is preserved. The result is also noted to imply the Ramsey property for the class OMBA_{Q_2} of naturally ordered finite measure algebras with dyadic-rational measures.
Significance. Resolving the stated open question supplies a homogeneous counterpart to the Graham-Rothschild theorem and yields the Ramsey property for OMBA_{Q_2}, a concrete advance for the study of Ramsey properties of measure algebras and the topological dynamics of their automorphism groups. The reduction is parameter-free once the classical theorem is invoked and makes the homogeneous statement a direct corollary rather than an independent combinatorial statement.
minor comments (2)
- The manuscript is a short note; a brief remark on the length of the chosen multiple n (relative to the Graham-Rothschild number) would help readers verify the divisibility chain without recomputing the classical bound.
- Notation (n)^k_hom is introduced in the abstract but could be restated once in the body before the reduction argument for self-contained reading.
Simulated Author's Rebuttal
We thank the referee for the careful reading and the positive report recommending acceptance. The summary accurately reflects the content and implications of the note.
Circularity Check
No significant circularity identified
full rationale
The paper answers an open question posed by Kechris-Sokic-Todorcevic by supplying an explicit reduction of the homogeneous Dual Ramsey statement to the classical (non-homogeneous) Graham-Rothschild Dual Ramsey theorem via a canonical extension of colorings that preserves the relevant monochromaticity properties. This reduction is constructed directly from the external theorem and does not rely on any self-definition, fitted parameters renamed as predictions, load-bearing self-citations, or ansatzes imported from the author's prior work. The derivation chain is therefore independent of the target result and self-contained against the cited external benchmark.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math The finite Dual Ramsey Theorem of Graham-Rothschild holds for ordinary (non-homogeneous) partitions.
discussion (0)
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