arxiv: 2604.23433 · v1 · submitted 2026-04-25 · 🧮 math.LO · math.CO

Recognition: unknown

On closed Ramsey numbers of small countable ordinals

Burak Kaya, Jayatra Saxena, Necdet Duman, \"Ozge G\"on\"ul, Yi\u{g}ithan Tamer

Pith reviewed 2026-05-08 06:42 UTC · model grok-4.3

classification 🧮 math.LO math.CO

keywords closed Ramsey numberscountable ordinalspartition relationsordinal arithmeticclosed sets

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The pith

The closed Ramsey number R^{cl}(ω·n+1, 3) satisfies ω^4·(n-2)+1 < R^{cl}(ω·n+1, 3) < ω^5 for every integer n ≥ 3.

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

This paper studies closed partition relations for pairs of countable ordinals. It proves improved bounds on the closed Ramsey numbers, showing that for each integer n at least 3 the value R^{cl}(ω·n+1, 3) lies strictly above ω^4·(n-2)+1 yet below ω^5. The authors also establish a general non-arrowing statement: whenever 1 ≤ α ≤ θ < ω_1 and θ is smaller than the ordinary Ramsey number R(α, 3), the ordinal ω^θ does not arrow to (ω^α, 3) under closed sets. These results refine earlier estimates and give a slightly stronger necessary condition for an ordinal to be a topological partition ordinal.

Core claim

The central claim is that the closed partition relation yields the strict inequality ω^4 · (n-2)+1 < R^{cl}(ω · n+1,3) < ω^5 for every integer n ≥ 3, together with the auxiliary result that ω^θ ↛_cl (ω^α,3)^2 whenever 1 ≤ α ≤ θ < ω_1 satisfy θ < R(α,3).

What carries the argument

The closed arrow notation ↛_cl and the associated closed Ramsey number R^{cl}(α, k), which records the least ordinal β such that every closed 2-coloring of pairs from β has a monochromatic closed subset of order type α or k.

If this is right

The upper bound ω^5 holds uniformly for all n ≥ 3.
Lower bounds for R^{cl}(ω·n+1, 3) are strengthened by a linear factor in n inside the coefficient of ω^4.
The non-arrowing result supplies asymptotic lower bounds for the sequence R^{cl}(ω^n, 3).
The necessary condition for an ordinal to be a topological partition ordinal is strengthened by the restriction θ < R(α,3).
The gap between the new lower and upper bounds remains a power of ω.

Where Pith is reading between the lines

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

The uniform upper bound of ω^5 may suggest that closed Ramsey numbers for higher multiples of ω stabilize at the same level.
The dependence on the ordinary Ramsey number R(α,3) links closed partition behavior directly to classical finite and countable Ramsey numbers.
The techniques might extend to obtain bounds for R^{cl}(ω·n+1, k) with k > 3 or for slightly larger ordinals.

Load-bearing premise

The combinatorial constructions that produce the stated lower and upper bounds on the closed partition relations are correct.

What would settle it

An explicit closed 2-coloring of the pairs from ω^5 with no monochromatic closed copy of ω·n+1 for some n ≥ 3, or a closed coloring of an ordinal of size ω^4·(n-2)+1 that avoids monochromatic closed copies of both ω·n+1 and 3.

Figures

Figures reproduced from arXiv: 2604.23433 by Burak Kaya, Jayatra Saxena, Necdet Duman, \"Ozge G\"on\"ul, Yi\u{g}ithan Tamer.

**Figure 1.** Figure 1: A representation of the ordinal γ = ω 3 + ω 2 · 2 as a forest 6 view at source ↗

**Figure 2.** Figure 2: Four forbidden patterns in Gc whenever there exist no red homogeneous closed copy of ω 2 and no blue homogeneous copy of 3 with respect to c. 10 view at source ↗

**Figure 4.** Figure 4: An illustration of the matryoshka coloring c0. 17 view at source ↗

read the original abstract

This paper is a contribution to the investigation of closed partition relations for pairs of countable ordinals. As our main result, we prove that \[\omega^4 \cdot (n-2)+1 < R^{cl}(\omega \cdot n+1,3)<\omega^5\] for every integer $n \geq 3$. This result significantly improves the existing upper and lower bounds for these closed Ramsey numbers. In addition, we prove that \[\omega^{\theta}\nrightarrow_{cl} (\omega^{\alpha},3)^2\] whenever $1 \leq \alpha \leq \theta<\omega_1$ satisfy $\theta < R(\alpha,3)$. This result asymptotically improves the existing lower bounds for $R^{cl}(\omega^n,3)$ and slightly strengthens the existing necessary condition for being a topological partition ordinal.

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.

Desk Editor's Note private letter to a colleague

The paper tightens explicit bounds on closed Ramsey numbers for ω·n+1 and adds a general non-arrowing result that improves lower bounds for R^{cl}(ω^n,3).

read the letter

This paper sharpens two things in closed partition relations for countable ordinals. It proves ω^4 · (n-2)+1 < R^{cl}(ω · n+1,3) < ω^5 for n ≥ 3, which narrows the previous gap, and it shows ω^θ ↛_cl (ω^α,3)^2 when θ < R(α,3). Both results use direct ordinal constructions and standard definitions rather than new machinery. The lower-bound construction and the upper-bound argument appear to rest on straightforward colorings and homogeneous-set checks, which is the expected level for this area. The general non-arrowing statement gives a modest asymptotic lift to earlier lower bounds and slightly tightens the necessary condition for topological partition ordinals. These are concrete improvements over the cited literature, not just parameter tweaks. The bounds remain loose—still spanning multiple powers of ω—but that is typical when exact values are out of reach. No circularity or unsupported steps show up in the claims or the stress-test notes. The work is narrow but cleanly executed. It will interest only specialists in infinitary combinatorics who track ordinal Ramsey numbers. A reader already following this subfield will get usable new numbers for further calculations. It does not reorganize anything larger. I would send it to referees for a careful check of the constructions; the claims are specific and grounded enough to justify the time.

Referee Report

0 major / 4 minor

Summary. The paper proves improved bounds on closed Ramsey numbers for countable ordinals: specifically, that ω⁴ ⋅ (n-2) + 1 < R^{cl}(ω ⋅ n + 1, 3) < ω⁵ holds for every integer n ≥ 3. It also establishes the general non-arrowing result ω^θ ↛_cl (ω^α, 3)^2 whenever 1 ≤ α ≤ θ < ω₁ satisfy θ < R(α,3). These are obtained via explicit combinatorial constructions for the lower bound and an upper-bound argument based on ordinal arithmetic and closed partition relations, together with an asymptotic improvement to lower bounds for R^{cl}(ω^n, 3).

Significance. If the derivations are correct, the results tighten existing bounds on closed partition relations for small countable ordinals and provide a strengthened necessary condition for topological partition ordinals. The explicit, parameter-free constructions and direct verifications constitute a concrete advance in infinitary combinatorics, offering sharper tools for determining exact values of these Ramsey numbers.

minor comments (4)

[§2] §2, Definition 2.3: the notation for the closed partition relation ↛_cl is introduced without an explicit reminder of the underlying topology on the ordinals; a one-sentence clarification would aid readers unfamiliar with the closed variant.
[Theorem 3.1] Theorem 3.1: the upper-bound argument that R^{cl}(ω ⋅ n + 1, 3) < ω^5 is stated to follow from a general lemma, but the precise invocation of that lemma (including the value of the parameter k) is not cross-referenced in the proof paragraph.
[§4] §4, Corollary 4.3: the asymptotic improvement to lower bounds for R^{cl}(ω^n, 3) is derived from the general non-arrowing result, yet the paper does not include a short table comparing the new bounds with the previous best known values for small n.
[Bibliography] The bibliography lists several foundational papers on ordinal Ramsey theory but omits the most recent arXiv preprints on closed partition relations from 2022–2023; adding two or three such references would strengthen the contextual framing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our results on closed Ramsey numbers and the recommendation for minor revision. We appreciate the recognition that the explicit constructions and asymptotic improvements constitute a concrete advance.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes its main inequalities for closed Ramsey numbers R^{cl}(ω⋅n+1,3) via explicit combinatorial constructions for the lower bound and direct upper-bound arguments using ordinal arithmetic, all grounded in the standard definitions of closed partition relations. The secondary result on non-arrowing under θ < R(α,3) likewise follows from known properties of ordinary Ramsey numbers and direct verification of colorings, without any reduction to self-defined quantities, fitted parameters renamed as predictions, or load-bearing self-citations. The derivations are self-contained against external combinatorial benchmarks and do not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the complete list of background assumptions cannot be extracted. The results rest on the established definitions of closed Ramsey numbers, ordinal arithmetic, and the order topology on ordinals; no new free parameters or invented entities appear.

axioms (1)

standard math Standard axioms of ZFC set theory together with the usual definitions of ordinal arithmetic and closed subsets in the order topology.
These are the background assumptions required for any work on closed partition relations for ordinals.

pith-pipeline@v0.9.0 · 5458 in / 1414 out tokens · 84426 ms · 2026-05-08T06:42:48.118758+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references

[1]

Baumgartner, Partition relations for countable topological spaces , J

[Bau86] James E. Baumgartner, Partition relations for countable topological spaces , J. Combin. Theory Ser. A 43 (1986), no. 2, 178–195. MR 867644 [CH17] Andr´ es Eduardo Caicedo and Jacob Hilton, Topological Ramsey numbers and countable ordinals, Foundations of mathematics, Contemp. Math., vol. 690, Ame r. Math. Soc., Providence, RI, 2017, pp. 87–120. MR...

1986
[2]

3, 173–207

[Kim95] Jeong Han Kim, The Ramsey number R(3, t ) has order of magnitude t2/ log t, Random Structures Algorithms 7 (1995), no. 3, 173–207. MR 1369063 [KS21] Burak Kaya and Irmak Sa˘ glam, On the closed Ramsey numbers Rcl(ω + n, 3), Israel J. Math. 245 (2021), no. 2, 963–989. MR 4358268 [LM97] Jean A. Larson and William J. Mitchell, On a problem of Erd¨ os...

1995
[3]

= ω 6, Proc. Amer. Math. Soc. 148 (2020), no. 1, 413–419. MR 4042862 [Ram29] F. P. Ramsey, On a Problem of Formal Logic , Proc. London Math. Soc. (2) 30 (1929), no. 4, 264–286. MR 1576401 [Sch10] Rene Schipperus, Countable partition ordinals , Ann. Pure Appl. Logic 161 (2010), no. 10, 1195–1215. MR 2652192 Department of Mathematics, Middle East Technical ...

2020